diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,29 @@ +Copyright (c) 2008-2016, Balazs Komuves +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are met: + +- Redistributions of source code must retain the above copyright notice, +this list of conditions and the following disclaimer. + +- Redistributions in binary form must reproduce the above copyright notice, +this list of conditions and the following disclaimer in the documentation +and/or other materials provided with the distribution. + +- Neither names of the copyright holders nor the names of the contributors +may be used to endorse or promote products derived from this software without +specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + diff --git a/Math/Combinat.hs b/Math/Combinat.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat.hs @@ -0,0 +1,76 @@ + +-- | A collection of functions to generate, manipulate, +-- visualize and count combinatorial objects like partitions, +-- compositions, permutations, braids, Young tableaux, +-- lattice paths, various tree structures, etc etc. +-- +-- +-- See also the @combinat-diagrams@ library for generating +-- graphical representations of (some of) these structure using +-- the @diagrams@ library (). +-- +-- +-- The long-term goals are +-- +-- (1) generate most of the standard structures; +-- +-- (2) manipulate these structures; +-- +-- (3) visualize these structures; +-- +-- (4) the generation should be efficient; +-- +-- (5) to be able to enumerate the structures +-- with constant memory usage; +-- +-- (6) to be able to randomly sample from them; +-- +-- (7) finally, to be a repository of algorithms. +-- +-- +-- The short-term goal is simply to generate +-- and manipulate many interesting structures. +-- +-- +-- Naming conventions (subject to change): +-- +-- * prime suffix: additional constrains, typically more general; +-- +-- * underscore prefix: use plain lists instead of other types with +-- enforced invariants; +-- +-- * \"random\" prefix: generates random objects +-- (typically with uniform distribution); +-- +-- * \"count\" prefix: counting functions. +-- +-- +-- This module re-exports the most commonly used modules. +-- + +module Math.Combinat + ( module Math.Combinat.Numbers + , module Math.Combinat.Sign + , module Math.Combinat.Sets + , module Math.Combinat.Tuples + , module Math.Combinat.Compositions + , module Math.Combinat.Partitions + , module Math.Combinat.Permutations + , module Math.Combinat.Tableaux + , module Math.Combinat.Trees + , module Math.Combinat.LatticePaths + , module Math.Combinat.ASCII + ) + where + +import Math.Combinat.Numbers +import Math.Combinat.Sign +import Math.Combinat.Sets +import Math.Combinat.Tuples +import Math.Combinat.Compositions +import Math.Combinat.Partitions +import Math.Combinat.Permutations +import Math.Combinat.Tableaux +import Math.Combinat.Trees +import Math.Combinat.LatticePaths +import Math.Combinat.ASCII diff --git a/Math/Combinat/ASCII.hs b/Math/Combinat/ASCII.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/ASCII.hs @@ -0,0 +1,438 @@ + +-- | A mini-DSL for ASCII drawing of structures. +-- +-- +-- From some structures there is also Graphviz and\/or @diagrams@ +-- () visualization support +-- (the latter in the separate libray @combinat-diagrams@). +-- + +module Math.Combinat.ASCII where + +-------------------------------------------------------------------------------- + +import Data.Char ( isSpace ) +import Data.List ( transpose , intercalate ) + +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * The basic ASCII type + +-- | The type of a (rectangular) ASCII figure. +-- Internally it is a list of lines of the same length plus the size. +-- +-- Note: The Show instance is pretty-printing, so that it\'s convenient in ghci. +-- +data ASCII = ASCII + { asciiSize :: (Int,Int) + , asciiLines :: [String] + } + +-- | A type class to have a simple way to draw things +class DrawASCII a where + ascii :: a -> ASCII + +instance Show ASCII where + show = asciiString + +-- | An empty (0x0) rectangle +emptyRect :: ASCII +emptyRect = ASCII (0,0) [] + +asciiXSize, asciiYSize :: ASCII -> Int +asciiXSize = fst . asciiSize +asciiYSize = snd . asciiSize + +asciiString :: ASCII -> String +asciiString (ASCII sz ls) = unlines ls + +printASCII :: ASCII -> IO () +printASCII = putStrLn . asciiString + +asciiFromLines :: [String] -> ASCII +asciiFromLines ls = ASCII (x,y) (map f ls) where + y = length ls + x = maximum (map length ls) + f l = l ++ replicate (x - length l) ' ' + +asciiFromString :: String -> ASCII +asciiFromString = asciiFromLines . lines + +-------------------------------------------------------------------------------- +-- * Alignment + +-- | Horizontal alignment +data HAlign + = HLeft + | HCenter + | HRight + deriving (Eq,Show) + +-- | Vertical alignment +data VAlign + = VTop + | VCenter + | VBottom + deriving (Eq,Show) + +data Alignment = Align HAlign VAlign + +-------------------------------------------------------------------------------- +-- * Separators + +-- | Horizontal separator +data HSep + = HSepEmpty -- ^ empty separator + | HSepSpaces Int -- ^ @n@ spaces + | HSepString String -- ^ some custom string, eg. @\" | \"@ + deriving Show + +hSepSize :: HSep -> Int +hSepSize hsep = case hsep of + HSepEmpty -> 0 + HSepSpaces k -> k + HSepString s -> length s + +hSepString :: HSep -> String +hSepString hsep = case hsep of + HSepEmpty -> "" + HSepSpaces k -> replicate k ' ' + HSepString s -> s + +-- | Vertical separator +data VSep + = VSepEmpty -- ^ empty separator + | VSepSpaces Int -- ^ @n@ spaces + | VSepString [Char] -- ^ some custom list of characters, eg. @\" - \"@ (the characters are interpreted as below each other) + deriving Show + +vSepSize :: VSep -> Int +vSepSize vsep = case vsep of + VSepEmpty -> 0 + VSepSpaces k -> k + VSepString s -> length s + +vSepString :: VSep -> [Char] +vSepString vsep = case vsep of + VSepEmpty -> [] + VSepSpaces k -> replicate k ' ' + VSepString s -> s + +-------------------------------------------------------------------------------- +-- * Concatenation + +-- | Horizontal append, centrally aligned, no separation. +(|||) :: ASCII -> ASCII -> ASCII +(|||) p q = hCatWith VCenter HSepEmpty [p,q] + +-- | Vertical append, centrally aligned, no separation. +(===) :: ASCII -> ASCII -> ASCII +(===) p q = vCatWith HCenter VSepEmpty [p,q] + +-- | Horizontal concatenation, top-aligned, no separation +hCatTop :: [ASCII] -> ASCII +hCatTop = hCatWith VTop HSepEmpty + +-- | Horizontal concatenation, bottom-aligned, no separation +hCatBot :: [ASCII] -> ASCII +hCatBot = hCatWith VBottom HSepEmpty + +-- | Vertical concatenation, left-aligned, no separation +vCatLeft :: [ASCII] -> ASCII +vCatLeft = vCatWith HLeft VSepEmpty + +-- | Vertical concatenation, right-aligned, no separation +vCatRight :: [ASCII] -> ASCII +vCatRight = vCatWith HRight VSepEmpty + +-- | General horizontal concatenation +hCatWith :: VAlign -> HSep -> [ASCII] -> ASCII +hCatWith valign hsep rects = ASCII (x',maxy) final where + n = length rects + maxy = maximum [ y | ASCII (_,y) _ <- rects ] + xsz = [ x | ASCII (x,_) _ <- rects ] + sep = hSepString hsep + sepx = length sep + rects1 = map (vExtendTo valign maxy) rects + x' = sum' xsz + (n-1)*sepx + final = map (intercalate sep) $ transpose (map asciiLines rects1) + +-- | General vertical concatenation +vCatWith :: HAlign -> VSep -> [ASCII] -> ASCII +vCatWith halign vsep rects = ASCII (maxx,y') final where + n = length rects + maxx = maximum [ x | ASCII (x,_) _ <- rects ] + ysz = [ y | ASCII (_,y) _ <- rects ] + sepy = vSepSize vsep + fullsep = transpose (replicate maxx $ vSepString vsep) :: [String] + rects1 = map (hExtendTo halign maxx) rects + y' = sum' ysz + (n-1)*sepy + final = intercalate fullsep $ map asciiLines rects1 + +-------------------------------------------------------------------------------- +-- * Padding + +-- | Horizontally pads with the given number of spaces, on both sides +hPad :: Int -> ASCII -> ASCII +hPad k (ASCII (x,y) ls) = ASCII (x+2*k,y) (map f ls) where + f l = pad ++ l ++ pad + pad = replicate k ' ' + +-- | Vertically pads with the given number of empty lines, on both sides +vPad :: Int -> ASCII -> ASCII +vPad k (ASCII (x,y) ls) = ASCII (x,y+2*k) (pad ++ ls ++ pad) where + pad = replicate k (replicate x ' ') + +-- | Pads by single empty lines vertically and two spaces horizontally +pad :: ASCII -> ASCII +pad = vPad 1 . hPad 2 + +-------------------------------------------------------------------------------- +-- * Extension + +-- | Extends an ASCII figure with spaces horizontally to the given width. +-- Note: the alignment is the alignment of the original picture in the new bigger picture! +hExtendTo :: HAlign -> Int -> ASCII -> ASCII +hExtendTo halign n0 rect@(ASCII (x,y) ls) = hExtendWith halign (max n0 x - x) rect + +-- | Extends an ASCII figure with spaces vertically to the given height. +-- Note: the alignment is the alignment of the original picture in the new bigger picture! +vExtendTo :: VAlign -> Int -> ASCII -> ASCII +vExtendTo valign n0 rect@(ASCII (x,y) ls) = vExtendWith valign (max n0 y - y) rect + +-- | Extend horizontally with the given number of spaces. +hExtendWith :: HAlign -> Int -> ASCII -> ASCII +hExtendWith alignment d (ASCII (x,y) ls) = ASCII (x+d,y) (map f ls) where + f l = case alignment of + HLeft -> l ++ replicate d ' ' + HRight -> replicate d ' ' ++ l + HCenter -> replicate a ' ' ++ l ++ replicate (d-a) ' ' + a = div d 2 + +-- | Extend vertically with the given number of empty lines. +vExtendWith :: VAlign -> Int -> ASCII -> ASCII +vExtendWith valign d (ASCII (x,y) ls) = ASCII (x,y+d) (f ls) where + f ls = case valign of + VTop -> ls ++ replicate d emptyline + VBottom -> replicate d emptyline ++ ls + VCenter -> replicate a emptyline ++ ls ++ replicate (d-a) emptyline + a = div d 2 + emptyline = replicate x ' ' + +-- | Horizontal indentation +hIndent :: Int -> ASCII -> ASCII +hIndent d = hExtendWith HRight d + +-- | Vertical indentation +vIndent :: Int -> ASCII -> ASCII +vIndent d = vExtendWith VBottom d + +-------------------------------------------------------------------------------- +-- * Cutting + +-- | Cuts the given number of columns from the picture. +-- The alignment is the alignment of the /picture/, not the cuts. +-- +-- This should be the (left) inverse of 'hExtendWith'. +hCut :: HAlign -> Int -> ASCII -> ASCII +hCut halign k (ASCII (x,y) ls) = ASCII (x',y) (map f ls) where + x' = max 0 (x-k) + f = case halign of + HLeft -> reverse . drop k . reverse + HCenter -> reverse . drop (k-a) . reverse . drop a + HRight -> drop k + a = div k 2 + +-- | Cuts the given number of rows from the picture. +-- The alignment is the alignment of the /picture/, not the cuts. +-- +-- This should be the (left) inverse of 'vExtendWith'. +vCut :: VAlign -> Int -> ASCII -> ASCII +vCut valign k (ASCII (x,y) ls) = ASCII (x,y') (g ls) where + y' = max 0 (y-k) + g = case valign of + VTop -> reverse . drop k . reverse + VCenter -> reverse . drop (k-a) . reverse . drop a + VBottom -> drop k + a = div k 2 + +-------------------------------------------------------------------------------- +-- * Pasting + +-- | Pastes the first ASCII graphics onto the second, keeping the second one's dimension +-- (that is, overlapping parts of the first one are ignored). +-- The offset is relative to the top-left corner of the second picture. +-- Spaces at treated as transparent. +-- +-- Example: +-- +-- > tabulate (HCenter,VCenter) (HSepSpaces 2, VSepSpaces 1) +-- > [ [ caption (show (x,y)) $ +-- > pasteOnto (x,y) (filledBox '@' (4,3)) (asciiBox (7,5)) +-- > | x <- [-4..7] ] +-- > | y <- [-3..5] ] +-- +pasteOnto :: (Int,Int) -> ASCII -> ASCII -> ASCII +pasteOnto = pasteOnto' isSpace + +-- | Pastes the first ASCII graphics onto the second, keeping the second one's dimension. +-- The first argument specifies the transparency condition (on the first picture). +-- The offset is relative to the top-left corner of the second picture. +-- +pasteOnto' + :: (Char -> Bool) -- ^ transparency condition + -> (Int,Int) -- ^ offset relative to the top-left corner of the second picture + -> ASCII -- ^ picture to paste + -> ASCII -- ^ picture to paste onto + -> ASCII +pasteOnto' transparent (xpos,ypos) small big = new where + new = ASCII (xbig,ybig) lines' + (xbig,ybig) = asciiSize big + bigLines = asciiLines big + small' = (if (ypos>=0) then vExtendWith VBottom ypos else vCut VBottom (-ypos)) + $ (if (xpos>=0) then hExtendWith HRight xpos else hCut HRight (-xpos)) + $ small + smallLines = asciiLines small' + lines' = zipWith f bigLines (smallLines ++ repeat "") + f bl sl = zipWith g bl (sl ++ repeat ' ') + g b s = if transparent s then b else s + +-- | A version of 'pasteOnto' where we can specify the corner of the second picture +-- to which the offset is relative: +-- +-- > pasteOntoRel (HLeft,VTop) == pasteOnto +-- +pasteOntoRel :: (HAlign,VAlign) -> (Int,Int) -> ASCII -> ASCII -> ASCII +pasteOntoRel = pasteOntoRel' isSpace + +pasteOntoRel' :: (Char -> Bool) -> (HAlign,VAlign) -> (Int,Int) -> ASCII -> ASCII -> ASCII +pasteOntoRel' transparent (halign,valign) (xpos,ypos) small big = new where + new = pasteOnto' transparent (xpos',ypos') small big + (xsize,ysize) = asciiSize big + xpos' = case halign of + HLeft -> xpos + HCenter -> xpos + div xsize 2 + HRight -> xpos + xsize + ypos' = case valign of + VTop -> ypos + VCenter -> ypos + div ysize 2 + VBottom -> ypos + ysize + +-------------------------------------------------------------------------------- +-- * Tabulate + +-- | Tabulates the given matrix of pictures. Example: +-- +-- > tabulate (HCenter, VCenter) (HSepSpaces 2, VSepSpaces 1) +-- > [ [ asciiFromLines [ "x=" ++ show x , "y=" ++ show y ] | x<-[7..13] ] +-- > | y<-[98..102] ] +-- +tabulate :: (HAlign,VAlign) -> (HSep,VSep) -> [[ASCII]] -> ASCII +tabulate (halign,valign) (hsep,vsep) rects0 = final where + n = length rects0 + m = maximum (map length rects0) + rects1 = map (\rs -> rs ++ replicate (m - length rs) emptyRect) rects0 + ys = map (\rs -> maximum (map asciiYSize rs)) rects1 + xs = map (\rs -> maximum (map asciiXSize rs)) (transpose rects1) + rects2 = map (\rs -> [ hExtendTo halign x r | (x,r ) <- zip xs rs ]) rects1 + rects3 = [ map (vExtendTo valign y) rs | (y,rs) <- zip ys rects2 ] + final = vCatWith HLeft vsep + $ map (hCatWith VTop hsep) rects3 + +-- | Order of elements in a matrix +data MatrixOrder + = RowMajor + | ColMajor + deriving (Eq,Ord,Show,Read) + +-- | Automatically tabulates ASCII rectangles. +-- +autoTabulate + :: MatrixOrder -- ^ whether to use row-major or column-major ordering of the elements + -> Either Int Int -- ^ @(Right x)@ creates x columns, while @(Left y)@ creates y rows + -> [ASCII] -- ^ list of ASCII rectangles + -> ASCII +autoTabulate mtxorder ei list = final where + + final = tabulate (HLeft,VBottom) (HSepSpaces 2,VSepSpaces 1) rects + + n = length list + + rects = case ei of + + Left y -> case mtxorder of + ColMajor -> transpose (parts y list) + RowMajor -> invparts y list + + Right x -> case mtxorder of + ColMajor -> transpose (invparts x list) + RowMajor -> parts x list + + transposeIf b = if b then transpose else id + + -- chops into parts (the last one can be smaller) + parts d = go where + go [] = [] + go xs = take d xs : go (drop d xs) + + invparts d xs = parts' ds xs where + (q,r) = divMod n d + ds = replicate r (q+1) ++ replicate (d-r) q + + parts' ds xs = go ds xs where + go _ [] = [] + go [] _ = [] + go (d:ds) xs = take d xs : go ds (drop d xs) + +-------------------------------------------------------------------------------- +-- * Captions + +-- | Adds a caption to the bottom, with default settings. +caption :: String -> ASCII -> ASCII +caption = caption' False HLeft + +-- | Adds a caption to the bottom. The @Bool@ flag specifies whether to add an empty between +-- the caption and the figure +caption' :: Bool -> HAlign -> String -> ASCII -> ASCII +caption' emptyline halign str rect = vCatWith halign sep [rect,capt] where + sep = if emptyline then VSepSpaces 1 else VSepEmpty + capt = asciiFromString str + +-------------------------------------------------------------------------------- +-- * Ready-made boxes + +-- | An ASCII border box of the given size +asciiBox :: (Int,Int) -> ASCII +asciiBox (x,y) = ASCII (max x 2, max y 2) (h : replicate (y-2) m ++ [h]) where + h = "+" ++ replicate (x-2) '-' ++ "+" + m = "|" ++ replicate (x-2) ' ' ++ "|" + +-- | An \"rounded\" ASCII border box of the given size +roundedAsciiBox :: (Int,Int) -> ASCII +roundedAsciiBox (x,y) = ASCII (max x 2, max y 2) (a : replicate (y-2) m ++ [b]) where + a = "/" ++ replicate (x-2) '-' ++ "\\" + m = "|" ++ replicate (x-2) ' ' ++ "|" + b = "\\" ++ replicate (x-2) '-' ++ "/" + +-- | A box simply filled with the given character +filledBox :: Char -> (Int,Int) -> ASCII +filledBox c (x0,y0) = asciiFromLines $ replicate y (replicate x c) where + x = max 0 x0 + y = max 0 y0 + +-- | A box of spaces +transparentBox :: (Int,Int) -> ASCII +transparentBox = filledBox ' ' + +-------------------------------------------------------------------------------- +-- * Testing \/ miscellanea + +-- | An integer +asciiNumber :: Int -> ASCII +asciiNumber = asciiShow + +asciiShow :: Show a => a -> ASCII +asciiShow = asciiFromLines . (:[]) . show + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Classes.hs b/Math/Combinat/Classes.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Classes.hs @@ -0,0 +1,66 @@ + +-- | Type classes for some common properties shared by different objects + +{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies #-} +module Math.Combinat.Classes where + +-------------------------------------------------------------------------------- + +-- | Emptyness +class CanBeEmpty a where + isEmpty :: a -> Bool + empty :: a + +-------------------------------------------------------------------------------- +-- * Partitions + +-- | Number of parts +class HasNumberOfParts a where + numberOfParts :: a -> Int + +-------------------------------------------------------------------------------- + +class HasWidth a where + width :: a -> Int + +class HasHeight a where + height :: a -> Int + +-------------------------------------------------------------------------------- + +-- | Weight (of partitions, tableaux, etc) +class HasWeight a where + weight :: a -> Int + +-------------------------------------------------------------------------------- + +-- | Duality (of partitions, tableaux, etc) +class HasDuality a where + dual :: a -> a + +-------------------------------------------------------------------------------- +-- * Tableau + +-- | Shape (of tableaux, skew tableaux) +class HasShape a s | a -> s where + shape :: a -> s + +-------------------------------------------------------------------------------- +-- * Trees + +-- | Number of nodes (of trees) +class HasNumberOfNodes t where + numberOfNodes :: t -> Int + +-- | Number of leaves (of trees) +class HasNumberOfLeaves t where + numberOfLeaves :: t -> Int + +-------------------------------------------------------------------------------- +-- * Permutations + +-- | Number of cycles (of partitions) +class HasNumberOfCycles p where + numberOfCycles :: p -> Int + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Compositions.hs b/Math/Combinat/Compositions.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Compositions.hs @@ -0,0 +1,109 @@ + +-- | Compositions. +-- +-- See eg. +-- + +module Math.Combinat.Compositions where + +-------------------------------------------------------------------------------- + +import System.Random + +import Math.Combinat.Sets ( randomChoice ) +import Math.Combinat.Numbers ( factorial , binomial ) +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * generating all compositions + +-- | A /composition/ of an integer @n@ into @k@ parts is an ordered @k@-tuple of nonnegative (sometimes positive) integers +-- whose sum is @n@. +type Composition = [Int] + +-- | Compositions fitting into a given shape and having a given degree. +-- The order is lexicographic, that is, +-- +-- > sort cs == cs where cs = compositions' shape k +-- +compositions' + :: [Int] -- ^ shape + -> Int -- ^ sum + -> [[Int]] +compositions' [] 0 = [[]] +compositions' [] _ = [] +compositions' shape@(s:ss) n = + [ x:xs | x <- [0..min s n] , xs <- compositions' ss (n-x) ] + +countCompositions' :: [Int] -> Int -> Integer +countCompositions' [] 0 = 1 +countCompositions' [] _ = 0 +countCompositions' shape@(s:ss) n = sum + [ countCompositions' ss (n-x) | x <- [0..min s n] ] + +-- | All positive compositions of a given number (filtrated by the length). +-- Total number of these is @2^(n-1)@ +allCompositions1 :: Int -> [[Composition]] +allCompositions1 n = map (\d -> compositions1 d n) [1..n] + +-- | All compositions fitting into a given shape. +allCompositions' :: [Int] -> [[Composition]] +allCompositions' shape = map (compositions' shape) [0..d] where d = sum shape + +-- | Nonnegative compositions of a given length. +compositions + :: Integral a + => a -- ^ length + -> a -- ^ sum + -> [[Int]] +compositions len' d' = compositions' (replicate len d) d where + len = fromIntegral len' + d = fromIntegral d' + +-- | # = \\binom { len+d-1 } { len-1 } +countCompositions :: Integral a => a -> a -> Integer +countCompositions len d = binomial (len+d-1) (len-1) + +-- | Positive compositions of a given length. +compositions1 + :: Integral a + => a -- ^ length + -> a -- ^ sum + -> [[Int]] +compositions1 len d + | len > d = [] + | otherwise = map plus1 $ compositions len (d-len) + where + plus1 = map (+1) + -- len = fromIntegral len' + -- d = fromIntegral d' + +countCompositions1 :: Integral a => a -> a -> Integer +countCompositions1 len d = countCompositions len (d-len) + +-------------------------------------------------------------------------------- +-- * random compositions + +-- | @randomComposition k n@ returns a uniformly random composition +-- of the number @n@ as an (ordered) sum of @k@ /nonnegative/ numbers +randomComposition :: RandomGen g => Int -> Int -> g -> ([Int],g) +randomComposition k n g0 = + if k<1 || n<0 + then error "randomComposition: k should be positive, and n should be nonnegative" + else (comp, g1) + where + (cs,g1) = randomChoice (k-1) (n+k-1) g0 + comp = pairsWith (\x y -> y-x-1) (0 : cs ++ [n+k]) + +-- | @randomComposition1 k n@ returns a uniformly random composition +-- of the number @n@ as an (ordered) sum of @k@ /positive/ numbers +randomComposition1 :: RandomGen g => Int -> Int -> g -> ([Int],g) +randomComposition1 k n g0 = + if k<1 || n +-- +-- +-- Based on: +-- +-- * Joan S. Birman, Tara E. Brendle: BRAIDS - A SURVEY +-- +-- +-- +-- Note: This module GHC 7.8, since we use type-level naturals +-- to parametrize the 'Braid' type. +-- + + +{-# LANGUAGE + CPP, BangPatterns, + ScopedTypeVariables, ExistentialQuantification, + DataKinds, KindSignatures, Rank2Types, + TypeOperators, TypeFamilies, + StandaloneDeriving #-} + +module Math.Combinat.Groups.Braid where + +-------------------------------------------------------------------------------- + +import Data.Proxy +import GHC.TypeLits + +import Control.Monad + +import Data.List ( mapAccumL , foldl' ) + +import Data.Array.Unboxed +import Data.Array.ST +import Data.Array.IArray +import Data.Array.MArray +import Data.Array.Unsafe +import Data.Array.Base + +import Control.Monad.ST + +import System.Random + +import Math.Combinat.ASCII +import Math.Combinat.Sign +import Math.Combinat.Helper +import Math.Combinat.TypeLevel +import Math.Combinat.Numbers.Series + +import Math.Combinat.Permutations ( Permutation(..) ) +import qualified Math.Combinat.Permutations as P + +-------------------------------------------------------------------------------- +-- * Artin generators + +-- | A standard Artin generator of a braid: @Sigma i@ represents twisting +-- the neighbour strands @i@ and @(i+1)@, such that strand @i@ goes /under/ strand @(i+1)@. +-- +-- Note: The strands are numbered @1..n@. +data BrGen + = Sigma !Int -- ^ @i@ goes under @(i+1)@ + | SigmaInv !Int -- ^ @i@ goes above @(i+1)@ + deriving (Eq,Ord,Show) + +-- | The strand (more precisely, the first of the two strands) the generator twistes +brGenIdx :: BrGen -> Int +brGenIdx g = case g of + Sigma i -> i + SigmaInv i -> i + +brGenSign :: BrGen -> Sign +brGenSign g = case g of + Sigma _ -> Plus + SigmaInv _ -> Minus + +brGenSignIdx :: BrGen -> (Sign,Int) +brGenSignIdx g = case g of + Sigma i -> (Plus ,i) + SigmaInv i -> (Minus,i) + +-- | The inverse of a braid generator +invBrGen :: BrGen -> BrGen +invBrGen g = case g of + Sigma i -> SigmaInv i + SigmaInv i -> Sigma i + +-------------------------------------------------------------------------------- +-- * The braid type + +-- | The braid group @B_n@ on @n@ strands. +-- The number @n@ is encoded as a type level natural in the type parameter. +-- +-- Braids are represented as words in the standard generators and their +-- inverses. +newtype Braid (n :: Nat) = Braid [BrGen] deriving (Show) + +-- | The number of strands in the braid +numberOfStrands :: KnownNat n => Braid n -> Int +numberOfStrands = fromInteger . natVal . braidProxy where + braidProxy :: Braid n -> Proxy n + braidProxy _ = Proxy + +-- | Sometimes we want to hide the type-level parameter @n@, for example when +-- dynamically creating braids whose size is known only at runtime. +data SomeBraid = forall n. KnownNat n => SomeBraid (Braid n) + +someBraid :: Int -> (forall (n :: Nat). KnownNat n => Braid n) -> SomeBraid +someBraid n polyBraid = + case snat of + SomeNat pxy -> SomeBraid (asProxyTypeOf1 polyBraid pxy) + where + snat = case someNatVal (fromIntegral n :: Integer) of + Just sn -> sn + Nothing -> error "someBraid: input is not a natural number" + +withSomeBraid :: SomeBraid -> (forall n. KnownNat n => Braid n -> a) -> a +withSomeBraid sbraid f = case sbraid of SomeBraid braid -> f braid + +mkBraid :: (forall n. KnownNat n => Braid n -> a) -> Int -> [BrGen] -> a +mkBraid f n w = y where + sb = someBraid n (Braid w) + y = withSomeBraid sb f + +withBraid + :: Int + -> (forall (n :: Nat). KnownNat n => Braid n) + -> (forall (n :: Nat). KnownNat n => Braid n -> a) + -> a +withBraid n polyBraid f = + case snat of + SomeNat pxy -> f (asProxyTypeOf1 polyBraid pxy) + where + snat = case someNatVal (fromIntegral n :: Integer) of + Just sn -> sn + Nothing -> error "withBraid: input is not a natural number" + +-------------------------------------------------------------------------------- + +braidWord :: Braid n -> [BrGen] +braidWord (Braid gs) = gs + +braidWordLength :: Braid n -> Int +braidWordLength (Braid gs) = length gs + +-- | Embeds a smaller braid group into a bigger braid group +extend :: (n1 <= n2) => Braid n1 -> Braid n2 +extend (Braid gs) = Braid gs + +-- | Apply \"free reduction\" to the word, that is, iteratively remove @sigma_i sigma_i^-1@ pairs. +-- The resulting braid is clearly equivalent to the original. +freeReduceBraidWord :: Braid n -> Braid n +freeReduceBraidWord (Braid orig) = Braid (loop orig) where + + loop w = case reduceStep w of + Nothing -> w + Just w' -> loop w' + + reduceStep :: [BrGen] -> Maybe [BrGen] + reduceStep = go False where + go !changed w = case w of + (Sigma x : SigmaInv y : rest) | x==y -> go True rest + (SigmaInv x : Sigma y : rest) | x==y -> go True rest + (this : rest) -> liftM (this:) $ go changed rest + _ -> if changed then Just w else Nothing + +-------------------------------------------------------------------------------- +-- * Some specific braids + +-- | The braid generator @sigma_i@ as a braid +sigma :: KnownNat n => Int -> Braid (n :: Nat) +sigma k = braid where + braid = if k > 0 && k < numberOfStrands braid + then Braid [Sigma k] + else error "sigma: braid generator index out of range" + +-- | The braid generator @sigma_i^(-1)@ as a braid +sigmaInv :: KnownNat n => Int -> Braid (n :: Nat) +sigmaInv k = braid where + braid = if k > 0 && k < numberOfStrands braid + then Braid [SigmaInv k] + else error "sigma: braid generator index out of range" + +-- | @doubleSigma s t@ (for s Int -> Int -> Braid (n :: Nat) +doubleSigma s t = braid where + n = numberOfStrands braid + braid + | s < 1 || s > n = error "doubleSigma: s index out of range" + | t < 1 || t > n = error "doubleSigma: t index out of range" + | s >= t = error "doubleSigma: s >= t" + | otherwise = Braid $ + [ Sigma i | i<-[t-1,t-2..s] ] ++ [ SigmaInv i | i<-[s+1..t-1] ] + +-- | @positiveWord [2,5,1]@ is shorthand for the word @sigma_2*sigma_5*sigma_1@. +positiveWord :: KnownNat n => [Int] -> Braid (n :: Nat) +positiveWord idxs = braid where + braid = Braid (map gen idxs) + n = numberOfStrands braid + gen i = if i>0 && i Braid n +halfTwist = braid where + braid = Braid $ map Sigma $ _halfTwist n + n = numberOfStrands braid + +-- | The untyped version of 'halfTwist' +_halfTwist :: Int -> [Int] +_halfTwist n = gens where + gens = concat [ sub k | k<-[1..n-1] ] + sub k = [ j | j<-[n-1,n-2..k] ] + +-- | Synonym for 'halfTwist' +theGarsideBraid :: KnownNat n => Braid n +theGarsideBraid = halfTwist + +-- | The inner automorphism defined by @tau(X) = Delta^-1 X Delta@, +-- where @Delta@ is the positive half-twist. +-- +-- This sends each generator @sigma_j@ to @sigma_(n-j)@. +-- +tau :: KnownNat n => Braid n -> Braid n +tau braid@(Braid gens) = Braid (map f gens) where + n = numberOfStrands braid + f (Sigma i) = Sigma (n-i) + f (SigmaInv i) = SigmaInv (n-i) + + +-- | The involution @tau@ on permutations (permutation braids) +-- +tauPerm :: Permutation -> Permutation +tauPerm (Permutation arr) = Permutation $ listArray (1,n) [ (n+1) - arr!(n-i) | i<-[0..n-1] ] where + (1,n) = bounds arr + +-------------------------------------------------------------------------------- +-- * Group operations + +-- | The trivial braid +identity :: Braid n +identity = Braid [] + +-- | The inverse of a braid. Note: we do not perform reduction here, +-- as a word is reduced if and only if its inverse is reduced. +inverse :: Braid n -> Braid n +inverse = Braid . reverse . map invBrGen . braidWord + +-- | Composes two braids, doing free reduction on the result +-- (that is, removing @(sigma_k * sigma_k^-1)@ pairs@) +compose :: Braid n -> Braid n -> Braid n +compose (Braid gs) (Braid hs) = freeReduceBraidWord $ Braid (gs++hs) + +composeMany :: [Braid n] -> Braid n +composeMany = freeReduceBraidWord . Braid . concat . map braidWord + +-- | Composes two braids without doing any reduction. +composeDontReduce :: Braid n -> Braid n -> Braid n +composeDontReduce (Braid gs) (Braid hs) = Braid (gs++hs) + +-------------------------------------------------------------------------------- +-- * Braid permutations + +-- | A braid is pure if its permutation is trivial +isPureBraid :: KnownNat n => Braid n -> Bool +isPureBraid braid = (braidPermutation braid == P.identity n) where + n = numberOfStrands braid + +-- | Returns the left-to-right permutation associated to the braid. +-- We follow the strands /from the left to the right/ (or from the top to the +-- bottom), and return the permutation taking the left side to the right side. +-- +-- This is compatible with /right/ (standard) action of the permutations: +-- @permuteRight (braidPermutationRight b1)@ corresponds to the left-to-right +-- permutation of the strands; also: +-- +-- > (braidPermutation b1) `multiply` (braidPermutation b2) == braidPermutation (b1 `compose` b2) +-- +-- Writing the right numbering of the strands below the left numbering, +-- we got the two-line notation of the permutation. +-- +braidPermutation :: KnownNat n => Braid n -> Permutation +braidPermutation braid@ (Braid gens) = perm where + n = numberOfStrands braid + perm = _braidPermutation n (map brGenIdx gens) + +-- | This is an untyped version of 'braidPermutation' +_braidPermutation :: Int -> [Int] -> Permutation +_braidPermutation n idxs = Permutation (runSTUArray action) where + + action :: forall s. ST s (STUArray s Int Int) + action = do + arr <- newArray_ (1,n) + forM_ [1..n] $ \i -> writeArray arr i i + worker arr idxs + return arr + + worker arr = go where + go [] = return arr + go (i:is) = do + a <- readArray arr i + b <- readArray arr (i+1) + writeArray arr i b + writeArray arr (i+1) a + go is + +-------------------------------------------------------------------------------- +-- * Permutation braids + +-- | A positive braid word contains only positive (@Sigma@) generators. +isPositiveBraidWord :: KnownNat n => Braid n -> Bool +isPositiveBraidWord (Braid gs) = all (isPlus . brGenSign) gs + +-- | A /permutation braid/ is a positive braid where any two strands cross +-- at most one, and /positively/. +-- +isPermutationBraid :: KnownNat n => Braid n -> Bool +isPermutationBraid braid = isPositiveBraidWord braid && crosses where + crosses = and [ check i j | i<-[1..n-1], j<-[i+1..n] ] + check i j = zeroOrOne (lkMatrix ! (i,j)) + zeroOrOne a = (a==1 || a==0) + lkMatrix = linkingMatrix braid + n = numberOfStrands braid + +-- | Untyped version of 'isPermutationBraid' for positive words. +_isPermutationBraid :: Int -> [Int] -> Bool +_isPermutationBraid n gens = crosses where + crosses = and [ check i j | i<-[1..n-1], j<-[i+1..n] ] + check i j = zeroOrOne (lkMatrix ! (i,j)) + zeroOrOne a = (a==1 || a==0) + lkMatrix = _linkingMatrix n $ map Sigma gens + +-- | For any permutation this functions returns a /permutation braid/ realizing +-- that permutation. Note that this is not unique, so we make an arbitrary choice +-- (except for the permutation @[n,n-1..1]@ reversing the order, in which case +-- the result must be the half-twist braid). +-- +-- The resulting braid word will have a length at most @choose n 2@ (and will have +-- that length only for the permutation @[n,n-1..1]@) +-- +-- > braidPermutationRight (permutationBraid perm) == perm +-- > isPermutationBraid (permutationBraid perm) == True +-- +permutationBraid :: KnownNat n => Permutation -> Braid n +permutationBraid perm = braid where + n1 = numberOfStrands braid + n2 = P.permutationSize perm + braid = if n1 == n2 + then Braid (map Sigma $ _permutationBraid perm) + else error $ "permutationBraid: incompatible n: " ++ show n1 ++ " vs. " ++ show n2 + +-- | Untyped version of 'permutationBraid' +_permutationBraid :: Permutation -> [Int] +_permutationBraid = concat . _permutationBraid' + +-- | Returns the individual \"phases\" of the a permutation braid realizing the +-- given permutation. +_permutationBraid' :: Permutation -> [[Int]] +_permutationBraid' perm@(Permutation arr) = runST action where + (1,n) = bounds arr + + action :: forall s. ST s [[Int]] + action = do + + -- cfwd = the current state of strands : cfwd!j = where is strand #j now? + -- cinv = the inverse of that permutation : cinv!i = which strand is on the #i position now? + + cfwd <- newArray_ (1,n) :: ST s (STUArray s Int Int) + cinv <- newArray_ (1,n) :: ST s (STUArray s Int Int) + forM_ [1..n] $ \j -> do + writeArray cfwd j j + writeArray cinv j j + + let doSwap i = do + a <- readArray cinv i + b <- readArray cinv (i+1) + writeArray cinv i b + writeArray cinv (i+1) a + + u <- readArray cfwd a + v <- readArray cfwd b + writeArray cfwd a v + writeArray cfwd b u + + -- at the k-th phase, we move the (inv!k)-th strand, which is the k-th strand /on the RHS/, to correct position. + let worker phase + | phase >= n = return [] + | otherwise = do + let tgt = (arr ! phase) + src <- readArray cfwd tgt + let this = [src-1,src-2..phase] + mapM_ doSwap $ this + rest <- worker (phase+1) + return (this:rest) + + worker 1 + + +-- | We compute the linking numbers between all pairs of strands: +-- +-- > linkingMatrix braid ! (i,j) == strandLinking braid i j +-- +linkingMatrix :: KnownNat n => Braid n -> UArray (Int,Int) Int +linkingMatrix braid@(Braid gens) = _linkingMatrix (numberOfStrands braid) gens where + +-- | Untyped version of 'linkingMatrix' +_linkingMatrix :: Int -> [BrGen] -> UArray (Int,Int) Int +_linkingMatrix n gens = runSTUArray action where + + action :: forall s. ST s (STUArray s (Int,Int) Int) + action = do + perm <- newArray_ (1,n) :: ST s (STUArray s Int Int) + forM_ [1..n] $ \i -> writeArray perm i i + let doSwap :: Int -> ST s () + doSwap i = do + a <- readArray perm i + b <- readArray perm (i+1) + writeArray perm i b + writeArray perm (i+1) a + + mat <- newArray ((1,1),(n,n)) 0 :: ST s (STUArray s (Int,Int) Int) + let doAdd :: Int -> Int -> Int -> ST s () + doAdd i j pm1 = do + x <- readArray mat (i,j) + writeArray mat (i,j) (x+pm1) + writeArray mat (j,i) (x+pm1) + + forM_ gens $ \g -> do + let (sgn,k) = brGenSignIdx g + u <- readArray perm k + v <- readArray perm (k+1) + doAdd u v (signValue sgn) + doSwap k + + return mat + + +-- | The linking number between two strands numbered @i@ and @j@ +-- (numbered such on the /left/ side). +strandLinking :: KnownNat n => Braid n -> Int -> Int -> Int +strandLinking braid@(Braid gens) i0 j0 + | i0 < 1 || i0 > n = error $ "strandLinkingNumber: invalid strand index i: " ++ show i0 + | j0 < 1 || j0 > n = error $ "strandLinkingNumber: invalid strand index j: " ++ show j0 + | i0 == j0 = 0 + | otherwise = go i0 j0 gens + where + n = numberOfStrands braid + + go !i !j [] = 0 + go !i !j (g:gs) + | i == k && j == k+1 = s + go (i+1) (j-1) gs + | j == k && i == k+1 = s + go (i-1) (j+1) gs + | i == k = go (i+1) j gs + | i == k+1 = go (i-1) j gs + | j == k = go i (j+1) gs + | j == k+1 = go i (j-1) gs + | otherwise = go i j gs + where + (sgn,k) = brGenSignIdx g + s = signValue sgn + +-------------------------------------------------------------------------------- +-- * Growth + +-- | Bronfman's recursive formula for the reciprocial of the growth function +-- of /positive/ braids. It was already known (by Deligne) that these generating functions +-- are reciprocials of polynomials; Bronfman [1] gave a recursive formula for them. +-- +-- > let count n l = length $ nub $ [ braidNormalForm w | w <- allPositiveBraidWords n l ] +-- > let convertPoly (1:cs) = zip (map negate cs) [1..] +-- > pseries' (convertPoly $ bronfmanH n) == expandBronfmanH n == [ count n l | l <- [0..] ] +-- +-- * [1] Aaron Bronfman: Growth functions of a class of monoids. Preprint, 2001 +-- +bronfmanH :: Int -> [Int] +bronfmanH n = bronfmanHsList !! n + +-- | An infinite list containing the Bronfman polynomials: +-- +-- > bronfmanH n = bronfmanHsList !! n +-- +bronfmanHsList :: [[Int]] +bronfmanHsList = list where + list = map go [0..] + go 0 = [1] + go n = sumSeries [ sgn i $ replicate (choose2 i) 0 ++ list !! (n-i) | i<-[1..n] ] + sgn i = if odd i then id else map negate + choose2 k = div (k*(k-1)) 2 + +-- | Expands the reciprocial of @H(n)@ into an infinite power series, +-- giving the growth function of the positive braids on @n@ strands. +expandBronfmanH :: Int -> [Int] +expandBronfmanH n = pseries' (convertPoly $ bronfmanH n) where + convertPoly (1:cs) = zip (map negate cs) [1..] + +-------------------------------------------------------------------------------- +-- * ASCII diagram + +instance KnownNat n => DrawASCII (Braid n) where + ascii = horizBraidASCII + +-- | Horizontal braid diagram, drawn from left to right, +-- with strands numbered from the bottom to the top +horizBraidASCII :: KnownNat n => Braid n -> ASCII +horizBraidASCII = horizBraidASCII' True + +-- | Horizontal braid diagram, drawn from left to right. +-- The boolean flag indicates whether to flip the strands +-- vertically ('True' means bottom-to-top, 'False' means top-to-bottom) +horizBraidASCII' :: KnownNat n => Bool -> Braid n -> ASCII +horizBraidASCII' flipped braid@(Braid gens) = final where + + n = numberOfStrands braid + + final = vExtendWith VTop 1 $ hCatTop allBlocks + allBlocks = prelude ++ middleBlocks ++ epilogue + prelude = [ numberBlock , spaceBlock , beginEndBlock ] + epilogue = [ beginEndBlock , spaceBlock , numberBlock' ] + middleBlocks = map block gens + + block g = case g of + Sigma i -> block' i $ if flipped then over else under + SigmaInv i -> block' i $ if flipped then under else over + + block' i middle = asciiFromLines $ drop 2 $ concat + $ replicate a horiz ++ [space3, middle] ++ replicate b horiz + where + (a,b) = if flipped then (n-i-1,i-1) else (i-1,n-i-1) + + -- cycleN :: Int -> [a] -> [a] + -- cycleN n = concat . replicate n + + spaceBlock = transparentBox (1,n*3-2) + beginEndBlock = asciiFromLines $ drop 2 $ concat $ replicate n horiz + numberBlock = mkNumbers [1..n] + numberBlock' = mkNumbers $ P.fromPermutation $ braidPermutation braid + + mkNumbers :: [Int] -> ASCII + mkNumbers list = vCatWith HRight (VSepSpaces 2) $ map asciiShow + $ (if flipped then reverse else id) $ list + + under = [ "\\ /" , " / " , "/ \\" ] + over = [ "\\ /" , " \\ " , "/ \\" ] + horiz = [ " " , " " , "___" ] + space3 = [ " " , " " , " " ] + +-------------------------------------------------------------------------------- + +{- this is unusably ugly and vertically loooong + +-- | Vertical braid diagram, drawn from the top to the bottom. +-- Strands are numbered from the left to the right. +-- +-- Writing down the strand numbers from the top and and the bottom +-- gives the two-line notation of the permutation realized by the braid. +-- +verticalBraidASCII :: KnownNat n => Braid n -> ASCII +verticalBraidASCII braid@(Braid gens) = final where + + n = numberOfStrands braid + + final = hExtendWith HLeft 1 $ vCatLeft allBlocks + allBlocks = prelude ++ middleBlocks ++ epilogue + prelude = [ numberBlock , spaceBlock , beginEndBlock ] + epilogue = [ beginEndBlock , spaceBlock , numberBlock' ] + middleBlocks = map block gens + + block g = case g of + Sigma i -> block' i under + SigmaInv i -> block' i over + + block' i middle = asciiFromLines (map f middle) where + f xs = drop 1 $ concat $ h (i-1) ++ [" ",xs] ++ h (n-i-1) + h k = replicate k " |" + + spaceBlock = transparentBox (n*3-2,1) + beginEndBlock = asciiFromLines $ replicate 3 $ drop 1 $ concat (replicate n " |") + numberBlock = mkNumbers [1..n] + numberBlock' = mkNumbers $ P.fromPermutation $ braidPermutation braid + + mkNumbers :: [Int] -> ASCII + mkNumbers list = asciiFromString (drop 1 $ concatMap show3 list) + show3 k = let s = show k + in replicate (3-length s) ' ' ++ s + + under = [ "\\ /" , " / " , "/ \\" ] + over = [ "\\ /" , " \\ " , "/ \\" ] + +-} + +-------------------------------------------------------------------------------- +-- * List of all words + +-- | All positive braid words of the given length +allPositiveBraidWords :: KnownNat n => Int -> [Braid n] +allPositiveBraidWords l = braids where + n = numberOfStrands (head braids) + braids = map Braid $ _allPositiveBraidWords n l + +-- | All braid words of the given length +allBraidWords :: KnownNat n => Int -> [Braid n] +allBraidWords l = braids where + n = numberOfStrands (head braids) + braids = map Braid $ _allBraidWords n l + +-- | Untyped version of 'allPositiveBraidWords' +_allPositiveBraidWords :: Int -> Int -> [[BrGen]] +_allPositiveBraidWords n = go where + go 0 = [[]] + go k = [ Sigma i : rest | i<-[1..n-1] , rest <- go (k-1) ] + +-- | Untyped version of 'allBraidWords' +_allBraidWords :: Int -> Int -> [[BrGen]] +_allBraidWords n = go where + go 0 = [[]] + go k = [ gen : rest | gen <- gens , rest <- go (k-1) ] + gens = concat [ [ Sigma i , SigmaInv i ] | i<-[1..n-1] ] + +-------------------------------------------------------------------------------- +-- * Random braids + +-- | Random braid word of the given length +randomBraidWord :: (RandomGen g, KnownNat n) => Int -> g -> (Braid n, g) +randomBraidWord len g = (braid, g') where + braid = Braid w + n = numberOfStrands braid + (w,g') = _randomBraidWord n len g + +-- | Random /positive/ braid word of the given length +randomPositiveBraidWord :: (RandomGen g, KnownNat n) => Int -> g -> (Braid n, g) +randomPositiveBraidWord len g = (braid, g') where + braid = Braid w + n = numberOfStrands braid + (w,g') = _randomPositiveBraidWord n len g + +-------------------------------------------------------------------------------- + +-- | Given a braid word, we perturb it randomly @m@ times using the braid relations, +-- so that the resulting new braid word is equivalent to the original. +-- +-- Useful for testing. +-- +randomPerturbBraidWord :: forall n g. (RandomGen g, KnownNat n) => Int -> Braid n -> g -> (Braid n, g) +randomPerturbBraidWord m braid@(Braid xs) g = (Braid word' , g') where + + (word',g') = go m (length xs) xs g + + n = numberOfStrands braid + + -- | A random pair cancelling each other + rndE :: g -> ([BrGen],g) + rndE g = (e1,g'') where + (i , g' ) = randomR (1,n-1) g + (b , g'' ) = random g' + e0 = [SigmaInv i, Sigma i] + e1 = if b then reverse e0 else e0 + + brg s i = case s of { Plus -> Sigma i ; Minus -> SigmaInv i } + brginv s i = case s of { Plus -> SigmaInv i ; Minus -> Sigma i } + + go :: Int -> Int -> [BrGen] -> g -> ([BrGen], g) + go !cnt !len !word !g + + | cnt <= 0 = (word, g) + + | len < 2 = let w' = if b1 then (e++word) else (word++e) -- if it is short, we just add a trivial pair somewhere + in continue g4 (len+2) w' + + | abs (i-j) >= 2 = continue g4 len (as ++ v:u:bs) -- they commute, so we just commute them + + | i == j && s/=t = continue g4 (len-2) (as ++ bs ) -- they are inverse of each other, so we kill them + + | abs (i-j) == 1 && s == t = let mid = if b1 + then [ brg s j , brg s i , brg s j , brginv s i ] -- insert pair and + else [ brginv s j , brg s i , brg s j , brg s i ] -- apply ternary relation + in continue g4 (len+2) (as ++ mid ++ bs) + + | otherwise = let mid = if b1 + then (u : e ++ [v]) + else if b2 + then [u,v] ++ e + else e ++ [u,v] + in continue g4 (len+2) (as++(u:e)++[v]++bs) -- otherwise we just insert an trivial pair + + where + + (pos , g1 ) = randomR (0,len-2) g + (b1 :: Bool , g2 ) = random g1 + (b2 :: Bool , g3 ) = random g2 + (e , g4 ) = rndE g3 + (as,u:v:bs) = splitAt pos word + (s,i) = brGenSignIdx u + (t,j) = brGenSignIdx v + + continue g' len' word' = go (cnt-1) len' word' g' + +-------------------------------------------------------------------------------- + +-- | This version of 'randomBraidWord' may be convenient to avoid the type level stuff +withRandomBraidWord + :: RandomGen g + => (forall n. KnownNat n => Braid n -> a) + -> Int -- ^ number of strands + -> Int -- ^ length of the random word + -> g -> (a, g) +withRandomBraidWord f n len = runRand $ do + withSelectedM f (rand $ randomBraidWord len) n + +-- | This version of 'randomPositiveBraidWord' may be convenient to avoid the type level stuff +withRandomPositiveBraidWord + :: RandomGen g + => (forall n. KnownNat n => Braid n -> a) + -> Int -- ^ number of strands + -> Int -- ^ length of the random word + -> g -> (a, g) +withRandomPositiveBraidWord f n len = runRand $ do + withSelectedM f (rand $ randomPositiveBraidWord len) n + +-- | Untyped version of 'randomBraidWord' +_randomBraidWord + :: (RandomGen g) + => Int -- ^ number of strands + -> Int -- ^ length of the random word + -> g -> ([BrGen], g) +_randomBraidWord n len = runRand $ replicateM len $ do + k <- randChoose (1,n-1) + s <- randRoll + return $ case s of + Plus -> Sigma k + Minus -> SigmaInv k + +-- | Untyped version of 'randomPositiveBraidWord' +_randomPositiveBraidWord + :: (RandomGen g) + => Int -- ^ number of strands + -> Int -- ^ length of the random word + -> g -> ([BrGen], g) +_randomPositiveBraidWord n len = runRand $ replicateM len $ do + liftM Sigma $ randChoose (1,n-1) + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Groups/Braid/NF.hs b/Math/Combinat/Groups/Braid/NF.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Groups/Braid/NF.hs @@ -0,0 +1,534 @@ + +-- | Normal form of braids, take 1. +-- +-- We implement the Adyan-Thurston-ElRifai-Morton solution to the word problem in braid groups. +-- +-- +-- Based on: +-- +-- * [1] Joan S. Birman, Tara E. Brendle: BRAIDS - A SURVEY +-- (chapter 5.1) +-- +-- * [2] Elsayed A. Elrifai, Hugh R. Morton: Algorithms for positive braids +-- + +{-# LANGUAGE + CPP, BangPatterns, + ScopedTypeVariables, ExistentialQuantification, + DataKinds, KindSignatures, Rank2Types #-} + +module Math.Combinat.Groups.Braid.NF + ( -- * Normal form + BraidNF (..) + , nfReprWord + , braidNormalForm + , braidNormalForm' + , braidNormalFormNaive' + -- * Starting and finishing sets + , permWordStartingSet + , permWordFinishingSet + , permutationStartingSet + , permutationFinishingSet + ) + where + +-------------------------------------------------------------------------------- + +import Data.Proxy +import GHC.TypeLits + +import Control.Monad + +import Data.List ( mapAccumL , foldl' , (\\) ) + +import Data.Array.Unboxed +import Data.Array.ST +import Data.Array.IArray +import Data.Array.MArray +import Data.Array.Unsafe +import Data.Array.Base + +import Control.Monad.ST + +import Math.Combinat.Helper +import Math.Combinat.Sign + +import Math.Combinat.Permutations ( Permutation(..) , isIdentityPermutation , isReversePermutation ) +import qualified Math.Combinat.Permutations as P + +import Math.Combinat.Groups.Braid + +-------------------------------------------------------------------------------- + +-- | A unique normal form for braids, called the /left-greedy normal form/. +-- It looks like @Delta^i*P@, where @Delta@ is the positive half-twist, @i@ is an integer, +-- and @P@ is a positive word, which can be further decomposed into non-@Delta@ /permutation words/; +-- these words themselves are not unique, but the permutations they realize /are/ unique. +-- +-- This will solve the word problem relatively fast, +-- though it is not the fastest known algorithm. +-- +data BraidNF (n :: Nat) = BraidNF + { _nfDeltaExp :: !Int -- ^ the exponent of @Delta@ + , _nfPerms :: [Permutation] -- ^ the permutations + } + deriving (Eq,Ord,Show) + +-- | A braid word representing the given normal form +nfReprWord :: KnownNat n => BraidNF n -> Braid n +nfReprWord (BraidNF k perms) = freeReduceBraidWord $ composeMany (deltas ++ rest) where + + deltas + | k > 0 = replicate k halfTwist + | k < 0 = replicate (-k) (inverse halfTwist) + | otherwise = [] + + rest = map permutationBraid perms + +-------------------------------------------------------------------------------- + +-- | Computes the normal form of a braid. We apply free reduction first, it should be faster that way. +braidNormalForm :: KnownNat n => Braid n -> BraidNF n +braidNormalForm = braidNormalForm' . freeReduceBraidWord + +-- | This function does not apply free reduction before computing the normal form +braidNormalForm' :: KnownNat n => Braid n -> BraidNF n +braidNormalForm' braid@(Braid gens) = BraidNF (dexp+pexp) perms where + n = numberOfStrands braid + invless = replaceInverses n gens + (dexp,posxword) = moveDeltasLeft n invless + factors = leftGreedyFactors n $ expandPosXWord n posxword + (pexp,perms) = normalizePermFactors n $ map (_braidPermutation n) factors + +-- | This one uses the naive inverse replacement method. Probably somewhat slower than 'braidNormalForm''. +braidNormalFormNaive' :: KnownNat n => Braid n -> BraidNF n +braidNormalFormNaive' braid@(Braid gens) = BraidNF (dexp+pexp) perms where + n = numberOfStrands braid + invless = replaceInversesNaive gens + (dexp,posxword) = moveDeltasLeft n invless + factors = leftGreedyFactors n $ expandPosXWord n posxword + (pexp,perms) = normalizePermFactors n $ map (_braidPermutation n) factors + +-------------------------------------------------------------------------------- + +-- | Replaces groups of @sigma_i^-1@ generators by @(Delta^-1 * P)@, +-- where @P@ is a positive word. +-- +-- This should be more clever (resulting in shorter words) than the naive version below +-- +replaceInverses :: Int -> [BrGen] -> [XGen] +replaceInverses n gens = worker gens where + + worker [] = [] + worker xs = replaceNegs neg ++ map (XSigma . brGenIdx) pos ++ worker rest where + (neg,tmp ) = span (isMinus . brGenSign) xs + (pos,rest) = span (isPlus . brGenSign) tmp + + replaceNegs gs = concatMap replaceFac facs where + facs = leftGreedyFactors n $ map brGenIdx gs + + replaceFac idxs = XDelta (-1) : map XSigma (_permutationBraid perm) where + perm = (P.reversePermutation n) `P.multiply` (P.adjacentTranspositions n idxs) + + +-- | Replaces @sigma_i^-1@ generators by @(Delta^-1 * L_i)@. +replaceInversesNaive :: [BrGen] -> [XGen] +replaceInversesNaive gens = concatMap f gens where + f (Sigma i) = [ XSigma i ] + f (SigmaInv i) = [ XDelta (-1) , XL i ] + +-------------------------------------------------------------------------------- + +-- | Temporary data structure to be used during the normal form computation +data XGen + = XDelta !Int -- ^ @Delta^k@ + | XSigma !Int -- ^ @Sigma_j@ + | XL !Int -- ^ @L_j = Delta * sigma_j^-1@ + | XTauL !Int -- ^ @tau(L_j)@ + deriving (Eq,Show) + +isXDelta :: XGen -> Bool +isXDelta x = case x of { XDelta {} -> True ; _ -> False } + +-- | We move the all @Delta@'s to the left +moveDeltasLeft :: Int -> [XGen] -> (Int,[XGen]) +moveDeltasLeft n input = (finalExp, finalPosWord) where + + (XDelta finalExp : finalPosWord) = reverse $ worker 0 (reverse input) + + -- we start from the right end, and work towards the left end + worker dexp [] = [ XDelta dexp ] + worker !dexp xs = this' ++ worker dexp' rest where + (delta,notdelta) = span isXDelta xs + (this ,rest ) = span (not . isXDelta) notdelta + dexp' = dexp + sumDeltas delta + this' = if even dexp' + then this + else map xtau this + + sumDeltas :: [XGen] -> Int + sumDeltas xs = foldl' (+) 0 [ k | XDelta k <- xs ] + + -- | The @X -> Delta^-1 * X * Delta@ inner automorphism + xtau :: XGen -> XGen + xtau (XSigma j) = XSigma (n-j) + xtau (XDelta k) = XDelta k + xtau (XL k) = XTauL k + xtau (XTauL k) = XL k + +-------------------------------------------------------------------------------- + +-- | Expands a /positive/ \"X-word\" into a positive braid word +expandPosXWord :: Int -> [XGen] -> [Int] +expandPosXWord n = concatMap f where + + posHalfTwist = _halfTwist n + + jtau :: Int -> Int + jtau j = n-j + + posLTable = listArray (1,n-1) [ _permutationBraid (posLPerm n i) | i<-[1..n-1] ] :: Array Int [Int] + posTauLTable = amap (map jtau) posLTable + + -- posRTable = listArray (1,n-1) [ _permutationBraid (posRPerm n i) | i<-[1..n-1] ] :: Array Int [Int] + + f x = case x of + XSigma i -> [i] + XL i -> posLTable ! i + XTauL i -> posTauLTable ! i + XDelta i + | i > 0 -> concat (replicate i posHalfTwist) + | i < 0 -> error "expandPosXWord: negative delta power" + | otherwise -> [] + + -- word :: Braid n -> [Int] + -- word (Braid gens) = map brGenIdx gens + + +-- | Expands an \"X-word\" into a braid word. Useful for debugging. +expandAnyXWord :: forall n. KnownNat n => [XGen] -> Braid n +expandAnyXWord xgens = braid where + n = numberOfStrands braid + + braid = composeMany (map f xgens) + + posHalfTwist = halfTwist :: Braid n + negHalfTwist = inverse posHalfTwist :: Braid n + + posLTable = listArray (1,n-1) [ permutationBraid (posLPerm n i) | i<-[1..n-1] ] :: Array Int (Braid n) + posTauLTable = amap tau posLTable + + -- posRTable = listArray (1,n-1) [ permutationBraid (posRPerm n i) | i<-[1..n-1] ] :: Array Int (Braid n) + + f :: XGen -> Braid n + f x = case x of + XSigma i -> sigma i + XL i -> posLTable ! i + XTauL i -> posTauLTable ! i + XDelta i + | i > 0 -> composeMany (replicate i posHalfTwist) + | i < 0 -> composeMany (replicate (-i) negHalfTwist) + | otherwise -> identity + +-------------------------------------------------------------------------------- + +-- | @posL k@ (denoted as @L_k@) is a /positive word/ which +-- satisfies @Delta = L_k * sigma_k@, or: +-- +-- > (inverse halfTwist) `compose` (posL k) ~=~ sigmaInv k@ +-- +-- Thus we can replace any word with a positive word plus some @Delta^-1@\'s +-- +posL :: KnownNat n => Int -> Braid n +posL k = braid where + n = numberOfStrands braid + braid = permutationBraid (posLPerm n k) + +-- | @posR k n@ (denoted as @R_k@) is a /permutation braid/ which +-- satisfies @Delta = sigma_k * R_k@ +-- +-- > (posR k) `compose` (inverse halfTwist) ~=~ sigmaInv k@ +-- +-- Thus we can replace any word with a positive word plus some @Delta^-1@'s +-- +posR :: KnownNat n => Int -> Braid n +posR k = braid where + n = numberOfStrands braid + braid = permutationBraid (posRPerm n k) + +-- | The permutation @posL k :: Braid n@ is realizing +posLPerm :: Int -> Int -> Permutation +posLPerm n k + | k>0 && k Int -> Permutation +posRPerm n k + | k>0 && k [[Int]] -> (Int, [[Int]]) +filterDeltaFactors n facs = (exp',facs'') where + + (exp',facs') = go 0 (reverse facs) + + jtau j = n-j + facs'' = reverse facs' + maxlen = div (n*(n-1)) 2 + + go !e [] = (e,[]) + go !e (xs:xxs) + | null xs = go e xxs + | length xs == maxlen = go (e+1) xxs + | otherwise = + if even e + then let (e',yys) = go e xxs in (e' , xs : yys) + else let (e',yys) = go e xxs in (e' , map jtau xs : yys) + +-------------------------------------------------------------------------------- + +-- | The /starting set/ of a positive braid P is the subset of @[1..n-1]@ defined by +-- +-- > S(P) = [ i | P = sigma_i * Q , Q is positive ] = [ i | (sigma_i^-1 * P) is positive ] +-- +-- This function returns the starting set a positive word, assuming it +-- is a /permutation braid/ (see Lemma 2.4 in [2]) +-- +permWordStartingSet :: Int -> [Int] -> [Int] +permWordStartingSet n xs = permWordFinishingSet n (reverse xs) + +-- | The /finishing set/ of a positive braid P is the subset of @[1..n-1]@ defined by +-- +-- > F(P) = [ i | P = Q * sigma_i , Q is positive ] = [ i | (P * sigma_i^-1) is positive ] +-- +-- This function returns the finishing set, assuming the input is a /permutation braid/ +-- +permWordFinishingSet :: Int -> [Int] -> [Int] +permWordFinishingSet n input = runST action where + + action :: forall s. ST s [Int] + action = do + perm <- newArray_ (1,n) :: ST s (STUArray s Int Int) + forM_ [1..n] $ \i -> writeArray perm i i + forM_ input $ \i -> do + a <- readArray perm i + b <- readArray perm (i+1) + writeArray perm i b + writeArray perm (i+1) a + flip filterM [1..n-1] $ \i -> do + a <- readArray perm i + b <- readArray perm (i+1) + return (b permutationStartingSet p == permWordStartingSet n (_permutationBraid p) +-- +permutationStartingSet :: Permutation -> [Int] +permutationStartingSet = permutationFinishingSet . P.inverse + +-- | This satisfies +-- +-- > permutationFinishingSet p == permWordFinishingSet n (_permutationBraid p) +-- +permutationFinishingSet :: Permutation -> [Int] +permutationFinishingSet (Permutation arr) + = [ i | i<-[1..n-1] , arr ! i > arr ! (i+1) ] where (1,n) = bounds arr + +-- | Returns the list of permutations failing Lemma 2.5 in [2] +-- (so an empty list means the implementaton is correct) +fails_lemmma_2_5 :: Int -> [Permutation] +fails_lemmma_2_5 n = [ p | p <- P.permutations n , not (test p) ] where + test p = and [ check i | i<-[1..n-1] ] where + w = _permutationBraid p + s = permWordStartingSet n w + check i = _isPermutationBraid n (i:w) == (not $ elem i s) + +-------------------------------------------------------------------------------- + +-- | Given factors defined as permutation braids, we normalize them +-- to /left-canonical form/ by ensuring that +-- +-- * for each consecutive pair @(P,Q)@ the finishing set F(P) contains the starting set S(Q) +-- +-- * all @Delta@-s (corresponding to the reverse permutation) are moved to the left +-- +-- * all trivial factors are filtered out +-- +-- Unfortunately, it seems that we may need multiple sweeps to do that... +-- +normalizePermFactors :: Int -> [Permutation] -> (Int,[Permutation]) +normalizePermFactors n = go 0 where + go !acc input = + if (exp==0 && input == output) + then (acc,input) + else go (acc+exp) output + where + (exp,output) = normalizePermFactors1 n input + +-- | Does 1 sweep of the above normalization process. +-- Unfortunately, it seems that we may need to do this multiple times... +-- +normalizePermFactors1 :: Int -> [Permutation] -> (Int,[Permutation]) +normalizePermFactors1 n input = (exp, reverse output) where + (exp, output) = worker 0 (reverse input) + + -- Notes: We work in reverse order, from the right to the left. + -- We maintain the number of Delta-s pushed through; the tau involutions + -- are implicit in the parity of this number + -- + worker :: Int -> [Permutation] -> (Int,[Permutation]) + worker = worker' 0 0 + + -- We also maintain additional 0/1 flip flags for the first two permutations + -- this is a little bit of hack but it should work nicely + -- + worker' :: Int -> Int -> Int -> [Permutation] -> (Int,[Permutation]) + worker' !ep !eq !e (!p : rest@(!q : rest')) + + -- check if the very first element is identity or Delta + -- (note: these are tau-invariants) + + | isIdentityPermutation p = worker' eq 0 e rest + | isReversePermutation p = worker' eq 0 (e+1) rest + + -- check if the second element is identity or Delta + -- this is necessary since we "fatten" the second element and it can possibly + -- become Delta after a while (?) + + | isIdentityPermutation q = worker' ep 0 e (p : rest') + | isReversePermutation q = worker' (ep-1) 0 (e+1) (p : rest') + + -- ok so we have something like "... : Q : P" + -- if F(Q) contains S(P) then we can move on; + -- otherwise there is an element j in S(P) \\ F(Q), so we can + -- replace it by "... : Qj : jP" + + | otherwise = + case permutationStartingSet preal \\ permutationFinishingSet qreal of + [] -> let (e',rs) = worker' eq 0 e rest in (e', preal : rs) + (j:_) -> worker' (-e) (-e) e (p':q':rest') where + s = P.adjacentTransposition n j + p' = P.multiply s preal + q' = P.multiply qreal s + where + preal = oddTau (e+ep) p -- the "real" p + qreal = oddTau (e+eq) q -- the "real" q + + worker' _ _ !e [ ] = (e,[]) + worker' !ep _ !e [p] + | isIdentityPermutation p = (e , []) + | isReversePermutation p = (e+1 , []) + | otherwise = (e , [oddTau (e+ep) p] ) + + oddTau :: Int -> Permutation -> Permutation + oddTau !e p = if even e then p else tauPerm p + +{- + checkDelta :: Int -> Permutation -> [Permutation] -> (Int,[Permutation]) + checkDelta !e !p !rest + | P.isIdentityPermutation p = worker e rest + | isReversePermutation p = worker (e+1) rest + | otherwise = let (e',rs) = worker e rest in (e', oddTau e p : rs) +-} + +-------------------------------------------------------------------------------- + +-- | Given a /positive/ word, we apply left-greedy factorization of +-- that word into subwords representing /permutation braids/. +-- +-- Example 5.1 from the above handbook: +-- +-- > leftGreedyFactors 7 [1,3,2,2,1,3,3,2,3,2] == [[1,3,2],[2,1,3],[3,2,3],[2]] +-- +leftGreedyFactors :: Int -> [Int] -> [[Int]] +leftGreedyFactors n input = filter (not . null) $ runST (action input) where + + action :: forall s. [Int] -> ST s [[Int]] + action input = do + + perm <- newArray_ (1,n) :: ST s (STUArray s Int Int) + forM_ [1..n] $ \i -> writeArray perm i i + let doSwap :: Int -> ST s () + doSwap i = do + a <- readArray perm i + b <- readArray perm (i+1) + writeArray perm i b + writeArray perm (i+1) a + + mat <- newArray ((1,1),(n,n)) 0 :: ST s (STUArray s (Int,Int) Int) + let clearMat = forM_ [1..n] $ \i -> + forM_ [1..n] $ \j -> writeArray mat (i,j) 0 + + let doAdd1 :: Int -> Int -> ST s Int + doAdd1 i j = do + x <- readArray mat (i,j) + let y = x+1 + writeArray mat (i,j) y + writeArray mat (j,i) y + return y + + let worker :: [Int] -> ST s [[Int]] + worker [] = return [[]] + worker (p:ps) = do + u <- readArray perm p + v <- readArray perm (p+1) + c <- doAdd1 u v + doSwap p + if c<=1 + then do + (f:fs) <- worker ps + return ((p:f):fs) + else do + clearMat + fs <- worker (p:ps) + return ([]:fs) + + worker input + +-------------------------------------------------------------------------------- + +{- + +-- | Finds ternary braid relations, and returns them as a list of indices, decorated +-- with a flag specifying which side of the relation we found, a sign specifying +-- whether it is a relation between positive or negative generators. +-- +findTernaryBraidRelations :: Braid n -> [(Int,Bool,Sign)] +findTernaryBraidRelations (Braid gens) = go 0 gens where + go !k (Sigma a : rest@(Sigma b : Sigma c : _)) + | a==c && b==a+1 = (k,True ,Plus) : go (k+1) rest + | a==c && b==a-1 = (k,False,Plus) : go (k+1) rest + | otherwise = go (k+1) rest + go !k (SigmaInv a : rest@(SigmaInv b : SigmaInv c : _)) + | a==c && b==a+1 = (k,True ,Minus) : go (k+1) rest + | a==c && b==a-1 = (k,False,Minus) : go (k+1) rest + | otherwise = go (k+1) rest + go !k (x:xs) = go (k+1) xs + go _ [] = [] + +-- | Finds subsequences like @(i,i+1,i)@ and @(i+1,i,i+1)@, and returns them +-- and a list of indices, plus a flag specifying which one we found (the first +-- one is 'True', second one is 'False') +-- +_findTernaryBraidRelations :: [Int] -> [(Int,Bool)] +_findTernaryBraidRelations = go 0 where + go !k (a:rest@(b:c:_)) + | a==c && b==a+1 = (k,True ) : go (k+1) rest + | a==c && b==a-1 = (k,False) : go (k+1) rest + | otherwise = go (k+1) rest + go !k (x:xs) = go (k+1) xs + go _ [] = [] + +-} + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Groups/Free.hs b/Math/Combinat/Groups/Free.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Groups/Free.hs @@ -0,0 +1,523 @@ + +-- | Words in free groups (and free powers of cyclic groups). +-- +-- This module is not re-exported by "Math.Combinat" +-- +{-# LANGUAGE CPP, BangPatterns, PatternGuards #-} +module Math.Combinat.Groups.Free where + +-------------------------------------------------------------------------------- + +-- new Base exports "Word" from Data.Word... +#ifdef MIN_VERSION_base +#if MIN_VERSION_base(4,7,1) +import Prelude hiding ( Word ) +#endif +#elif __GLASGOW_HASKELL__ >= 709 +import Prelude hiding ( Word ) +#endif + +import Data.Char ( chr ) +import Data.List ( mapAccumL , groupBy ) + +import Control.Monad ( liftM ) +import System.Random + +import Math.Combinat.Numbers +import Math.Combinat.Sign +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * Words + +-- | A generator of a (free) group, indexed by which \"copy\" of the group we are dealing with. +data Generator idx + = Gen !idx -- @a@ + | Inv !idx -- @a^(-1)@ + deriving (Eq,Ord,Show,Read) + +-- | The index of a generator +genIdx :: Generator idx -> idx +genIdx g = case g of + Gen x -> x + Inv x -> x + +-- | The sign of the (exponent of the) generator (that is, the generator is 'Plus', the inverse is 'Minus') +genSign :: Generator idx -> Sign +genSign g = case g of { Gen _ -> Plus ; Inv _ -> Minus } + +genSignValue :: Generator idx -> Int +genSignValue g = case g of { Gen _ -> (1::Int) ; Inv _ -> (-1::Int) } + +-- | keep the index, but return always the 'Gen' one. +absGen :: Generator idx -> Generator idx +absGen g = case g of + Gen x -> Gen x + Inv x -> Gen x + +-- | A /word/, describing (non-uniquely) an element of a group. +-- The identity element is represented (among others) by the empty word. +type Word idx = [Generator idx] + +-------------------------------------------------------------------------------- + +-- | Generators are shown as small letters: @a@, @b@, @c@, ... +-- and their inverses are shown as capital letters, so @A=a^-1@, @B=b^-1@, etc. +showGen :: Generator Int -> Char +showGen (Gen i) = chr (96+i) +showGen (Inv i) = chr (64+i) + +showWord :: Word Int -> String +showWord = map showGen + +-------------------------------------------------------------------------------- + +instance Functor Generator where + fmap f g = case g of + Gen x -> Gen (f x) + Inv y -> Inv (f y) + +-------------------------------------------------------------------------------- + +-- | The inverse of a generator +inverseGen :: Generator a -> Generator a +inverseGen g = case g of + Gen x -> Inv x + Inv x -> Gen x + +-- | The inverse of a word +inverseWord :: Word a -> Word a +inverseWord = map inverseGen . reverse + +-- | Lists all words of the given length (total number will be @(2g)^n@). +-- The numbering of the generators is @[1..g]@. +allWords + :: Int -- ^ @g@ = number of generators + -> Int -- ^ @n@ = length of the word + -> [Word Int] +allWords g = go where + go !0 = [[]] + go !n = [ x:xs | xs <- go (n-1) , x <- elems ] + elems = [ Gen a | a<-[1..g] ] + ++ [ Inv a | a<-[1..g] ] + +-- | Lists all words of the given length which do not contain inverse generators +-- (total number will be @g^n@). +-- The numbering of the generators is @[1..g]@. +allWordsNoInv + :: Int -- ^ @g@ = number of generators + -> Int -- ^ @n@ = length of the word + -> [Word Int] +allWordsNoInv g = go where + go !0 = [[]] + go !n = [ x:xs | xs <- go (n-1) , x <- elems ] + elems = [ Gen a | a<-[1..g] ] + +-------------------------------------------------------------------------------- +-- * Random words + +-- | A random group generator (or its inverse) between @1@ and @g@ +randomGenerator + :: RandomGen g + => Int -- ^ @g@ = number of generators + -> g -> (Generator Int, g) +randomGenerator !d !g0 = (gen, g2) where + (b, !g1) = random g0 + (k, !g2) = randomR (1,d) g1 + gen = if b then Gen k else Inv k + +-- | A random group generator (but never its inverse) between @1@ and @g@ +randomGeneratorNoInv + :: RandomGen g + => Int -- ^ @g@ = number of generators + -> g -> (Generator Int, g) +randomGeneratorNoInv !d !g0 = (Gen k, g1) where + (!k, !g1) = randomR (1,d) g0 + +-- | A random word of length @n@ using @g@ generators (or their inverses) +randomWord + :: RandomGen g + => Int -- ^ @g@ = number of generators + -> Int -- ^ @n@ = length of the word + -> g -> (Word Int, g) +randomWord !d !n !g0 = (word,g1) where + (g1,word) = mapAccumL (\g _ -> swap (randomGenerator d g)) g0 [1..n] + +-- | A random word of length @n@ using @g@ generators (but not their inverses) +randomWordNoInv + :: RandomGen g + => Int -- ^ @g@ = number of generators + -> Int -- ^ @n@ = length of the word + -> g -> (Word Int, g) +randomWordNoInv !d !n !g0 = (word,g1) where + (g1,word) = mapAccumL (\g _ -> swap (randomGeneratorNoInv d g)) g0 [1..n] + +-------------------------------------------------------------------------------- +-- * The free group on @g@ generators + +{-# SPECIALIZE multiplyFree :: Word Int -> Word Int -> Word Int #-} +{-# SPECIALIZE equivalentFree :: Word Int -> Word Int -> Bool #-} +{-# SPECIALIZE reduceWordFree :: Word Int -> Word Int #-} +{-# SPECIALIZE reduceWordFreeNaive :: Word Int -> Word Int #-} + +-- | Multiplication of the free group (returns the reduced result). It is true +-- for any two words w1 and w2 that +-- +-- > multiplyFree (reduceWordFree w1) (reduceWord w2) = multiplyFree w1 w2 +-- +multiplyFree :: Eq idx => Word idx -> Word idx -> Word idx +multiplyFree w1 w2 = reduceWordFree (w1 ++ w2) + +-- | Decides whether two words represent the same group element in the free group +equivalentFree :: Eq idx => Word idx -> Word idx -> Bool +equivalentFree w1 w2 = null $ reduceWordFree $ w1 ++ inverseWord w2 + +-- | Reduces a word in a free group by repeatedly removing @x*x^(-1)@ and +-- @x^(-1)*x@ pairs. The set of /reduced words/ forms the free group; the +-- multiplication is obtained by concatenation followed by reduction. +-- +reduceWordFree :: Eq idx => Word idx -> Word idx +reduceWordFree = loop where + + loop w = case reduceStep w of + Nothing -> w + Just w' -> loop w' + + reduceStep :: Eq a => Word a -> Maybe (Word a) + reduceStep = go False where + go !changed w = case w of + (Gen x : Inv y : rest) | x==y -> go True rest + (Inv x : Gen y : rest) | x==y -> go True rest + (this : rest) -> liftM (this:) $ go changed rest + _ -> if changed then Just w else Nothing + + +-- | Naive (but canonical) reduction algorithm for the free groups +reduceWordFreeNaive :: Eq idx => Word idx -> Word idx +reduceWordFreeNaive = loop where + loop w = let w' = step w in if w/=w' then loop w' else w + step = concatMap worker . groupBy (equating genIdx) where + worker gs + | s>0 = replicate s (Gen i) + | s<0 = replicate (abs s) (Inv i) + | otherwise = [] + where + i = genIdx (head gs) + s = sum' (map genSignValue gs) + +-------------------------------------------------------------------------------- + +-- | Counts the number of words of length @n@ which reduce to the identity element. +-- +-- Generating function is @Gf_g(u) = \\frac {2g-1} { g-1 + g \\sqrt{ 1 - (8g-4)u^2 } }@ +-- +countIdentityWordsFree + :: Int -- ^ g = number of generators in the free group + -> Int -- ^ n = length of the unreduced word + -> Integer +countIdentityWordsFree g n = countWordReductionsFree g n 0 + +-- | Counts the number of words of length @n@ whose reduced form has length @k@ +-- (clearly @n@ and @k@ must have the same parity for this to be nonzero): +-- +-- > countWordReductionsFree g n k == sum [ 1 | w <- allWords g n, k == length (reduceWordFree w) ] +-- +countWordReductionsFree + :: Int -- ^ g = number of generators in the free group + -> Int -- ^ n = length of the unreduced word + -> Int -- ^ k = length of the reduced word + -> Integer +countWordReductionsFree gens_ nn_ kk_ + | nn==0 = if k==0 then 1 else 0 + | even nn && kk == 0 = sum [ ( binomial (nn-i) (n -i) * gg^(i ) * (gg-1)^(n -i ) * ( i) ) `div` (nn-i) | i<-[0..n ] ] + | even nn && even kk = sum [ ( binomial (nn-i) (n-k-i) * gg^(i+1) * (gg-1)^(n+k-i-1) * (kk+i) ) `div` (nn-i) | i<-[0..n-k] ] + | odd nn && odd kk = sum [ ( binomial (nn-i) (n-k-i) * gg^(i+1) * (gg-1)^(n+k-i ) * (kk+i) ) `div` (nn-i) | i<-[0..n-k] ] + | otherwise = 0 + where + g = fromIntegral gens_ :: Integer + nn = fromIntegral nn_ :: Integer + kk = fromIntegral kk_ :: Integer + + gg = 2*g + n = div nn 2 + k = div kk 2 + +-------------------------------------------------------------------------------- +-- * Free powers of cyclic groups + +{-# SPECIALIZE multiplyZ2 :: Word Int -> Word Int -> Word Int #-} +{-# SPECIALIZE multiplyZ3 :: Word Int -> Word Int -> Word Int #-} +{-# SPECIALIZE multiplyZm :: Int -> Word Int -> Word Int -> Word Int #-} + +-- | Multiplication in free products of Z2's +multiplyZ2 :: Eq idx => Word idx -> Word idx -> Word idx +multiplyZ2 w1 w2 = reduceWordZ2 (w1 ++ w2) + +-- | Multiplication in free products of Z3's +multiplyZ3 :: Eq idx => Word idx -> Word idx -> Word idx +multiplyZ3 w1 w2 = reduceWordZ3 (w1 ++ w2) + +-- | Multiplication in free products of Zm's +multiplyZm :: Eq idx => Int -> Word idx -> Word idx -> Word idx +multiplyZm k w1 w2 = reduceWordZm k (w1 ++ w2) + +-------------------------------------------------------------------------------- + +{-# SPECIALIZE equivalentZ2 :: Word Int -> Word Int -> Bool #-} +{-# SPECIALIZE equivalentZ3 :: Word Int -> Word Int -> Bool #-} +{-# SPECIALIZE equivalentZm :: Int -> Word Int -> Word Int -> Bool #-} + +-- | Decides whether two words represent the same group element in free products of Z2 +equivalentZ2 :: Eq idx => Word idx -> Word idx -> Bool +equivalentZ2 w1 w2 = null $ reduceWordZ2 $ w1 ++ inverseWord w2 + +-- | Decides whether two words represent the same group element in free products of Z3 +equivalentZ3 :: Eq idx => Word idx -> Word idx -> Bool +equivalentZ3 w1 w2 = null $ reduceWordZ3 $ w1 ++ inverseWord w2 + +-- | Decides whether two words represent the same group element in free products of Zm +equivalentZm :: Eq idx => Int -> Word idx -> Word idx -> Bool +equivalentZm m w1 w2 = null $ reduceWordZm m $ w1 ++ inverseWord w2 + +-------------------------------------------------------------------------------- + +{-# SPECIALIZE reduceWordZ2 :: Word Int -> Word Int #-} +{-# SPECIALIZE reduceWordZ3 :: Word Int -> Word Int #-} +{-# SPECIALIZE reduceWordZm :: Int -> Word Int -> Word Int #-} + +-------------------------------------------------------------------------------- + +-- | Reduces a word, where each generator @x@ satisfies the additional relation @x^2=1@ +-- (that is, free products of Z2's) +reduceWordZ2 :: Eq idx => Word idx -> Word idx +reduceWordZ2 = loop where + loop w = case reduceStep w of + Nothing -> w + Just w' -> loop w' + + reduceStep :: Eq a => Word a -> Maybe (Word a) + reduceStep = go False where + go !changed w = case w of + (Gen x : Gen y : rest) | x==y -> go True rest + (Gen x : Inv y : rest) | x==y -> go True rest + (Inv x : Gen y : rest) | x==y -> go True rest + (Inv x : Inv y : rest) | x==y -> go True rest + (this : rest) -> liftM (absGen this:) $ go changed rest + _ -> if changed then Just w else Nothing + +-- | Reduces a word, where each generator @x@ satisfies the additional relation @x^3=1@ +-- (that is, free products of Z3's) +reduceWordZ3 :: Eq idx => Word idx -> Word idx +reduceWordZ3 = loop where + loop w = case reduceStep w of + Nothing -> w + Just w' -> loop w' + + reduceStep :: Eq a => Word a -> Maybe (Word a) + reduceStep = go False where + go !changed w = case w of + (Gen x : Inv y : rest) | x==y -> go True rest + (Inv x : Gen y : rest) | x==y -> go True rest + (Gen x : Gen y : Gen z : rest) | x==y && y==z -> go True rest + (Inv x : Inv y : Inv z : rest) | x==y && y==z -> go True rest + (Gen x : Gen y : rest) | x==y -> go True (Inv x : rest) -- !!! + (Inv x : Inv y : rest) | x==y -> go True (Gen x : rest) + (this : rest) -> liftM (this:) $ go changed rest + _ -> if changed then Just w else Nothing + +-- | Reduces a word, where each generator @x@ satisfies the additional relation @x^m=1@ +-- (that is, free products of Zm's) +reduceWordZm :: Eq idx => Int -> Word idx -> Word idx +reduceWordZm m = loop where + + loop w = case reduceStep w of + Nothing -> w + Just w' -> loop w' + + halfm = div m 2 -- if we encounter strictly more than m/2 equal elements in a row, we replace them by the inverses + + -- reduceStep :: Eq a => Word a -> Maybe (Word a) + reduceStep = go False where + go !changed w = case w of + (Gen x : Inv y : rest) | x==y -> go True rest + (Inv x : Gen y : rest) | x==y -> go True rest + something | Just (k,rest) <- dropIfMoreThanHalf w -> go True (replicate (m-k) (inverseGen (head w)) ++ rest) + (this : rest) -> liftM (this:) $ go changed rest + _ -> if changed then Just w else Nothing + + -- dropIfMoreThanHalf :: Eq a => Word a -> Maybe (Int, Word a) + dropIfMoreThanHalf w = + let (!k,rest) = dropWhileEqual w + in if k > halfm then Just (k,rest) + else Nothing + + -- dropWhileEqual :: Eq a => Word a -> (Int, Word a) + dropWhileEqual [] = (0,[]) + dropWhileEqual (x0:rest) = go 1 rest where + go !k [] = (k,[]) + go !k xxs@(x:xs) = if k==m then (m,xxs) + else if x==x0 then go (k+1) xs + else (k,xxs) + +{- + dropm :: Eq a => Word a -> Maybe (Word a) + dropm [] = Nothing + dropm (x:xs) = go (m-1) xs where + go 0 rest = Just rest + go j (y:ys) = if y==x + then go (j-1) ys + else Nothing + go j [] = Nothing +-} + +-------------------------------------------------------------------------------- + +{-# SPECIALIZE reduceWordZ2Naive :: Word Int -> Word Int #-} +{-# SPECIALIZE reduceWordZ3Naive :: Word Int -> Word Int #-} +{-# SPECIALIZE reduceWordZmNaive :: Int -> Word Int -> Word Int #-} + +-- | Reduces a word, where each generator @x@ satisfies the additional relation @x^2=1@ +-- (that is, free products of Z2's). Naive (but canonical) algorithm. +reduceWordZ2Naive :: Eq idx => Word idx -> Word idx +reduceWordZ2Naive = loop where + loop w = let w' = step w in if w/=w' then loop w' else w + step = concatMap worker . groupBy (equating genIdx) where + worker gs = + case mod s 2 of + 1 -> [Gen i] + 0 -> [] + _ -> error "reduceWordZ2: fatal error, shouldn't happen" + where + i = genIdx (head gs) + s = sum' (map genSignValue gs) + +-- | Reduces a word, where each generator @x@ satisfies the additional relation @x^3=1@ +-- (that is, free products of Z3's). Naive (but canonical) algorithm. +reduceWordZ3Naive :: Eq idx => Word idx -> Word idx +reduceWordZ3Naive = loop where + loop w = let w' = step w in if w/=w' then loop w' else w + step = concatMap worker . groupBy (equating genIdx) where + worker gs = + case mod s 3 of + 0 -> [] + 1 -> [Gen i] + 2 -> [Inv i] + _ -> error "reduceWordZ3: fatal error, shouldn't happen" + where + i = genIdx (head gs) + s = sum' (map genSignValue gs) + +-- | Reduces a word, where each generator @x@ satisfies the additional relation @x^m=1@ +-- (that is, free products of Zm's). Naive (but canonical) algorithm. +reduceWordZmNaive :: Eq idx => Int -> Word idx -> Word idx +reduceWordZmNaive m = loop where + loop w = let w' = step w in if w/=w' then loop w' else w + step = concatMap worker . groupBy (equating genIdx) where + halfm1 = div (m+1) 2 + worker gs + | mods <= halfm1 = replicate mods (Gen i) + | otherwise = replicate (m-mods) (Inv i) + where + i = genIdx (head gs) + s = sum' (map genSignValue gs) + mods = mod s m + +-------------------------------------------------------------------------------- + +-- | Counts the number of words (without inverse generators) of length @n@ +-- which reduce to the identity element, using the relations @x^2=1@. +-- +-- Generating function is @Gf_g(u) = \\frac {2g-2} { g-2 + g \\sqrt{ 1 - (4g-4)u^2 } }@ +-- +-- The first few @g@ cases: +-- +-- > A000984 = [ countIdentityWordsZ2 2 (2*n) | n<-[0..] ] = [1,2,6,20,70,252,924,3432,12870,48620,184756...] +-- > A089022 = [ countIdentityWordsZ2 3 (2*n) | n<-[0..] ] = [1,3,15,87,543,3543,23823,163719,1143999,8099511,57959535...] +-- > A035610 = [ countIdentityWordsZ2 4 (2*n) | n<-[0..] ] = [1,4,28,232,2092,19864,195352,1970896,20275660,211823800,2240795848...] +-- > A130976 = [ countIdentityWordsZ2 5 (2*n) | n<-[0..] ] = [1,5,45,485,5725,71445,925965,12335685,167817405,2321105525,32536755565...] +-- +countIdentityWordsZ2 + :: Int -- ^ g = number of generators in the free group + -> Int -- ^ n = length of the unreduced word + -> Integer +countIdentityWordsZ2 g n = countWordReductionsZ2 g n 0 + +-- | Counts the number of words (without inverse generators) of length @n@ whose +-- reduced form in the product of Z2-s (that is, for each generator @x@ we have @x^2=1@) +-- has length @k@ +-- (clearly @n@ and @k@ must have the same parity for this to be nonzero): +-- +-- > countWordReductionsZ2 g n k == sum [ 1 | w <- allWordsNoInv g n, k == length (reduceWordZ2 w) ] +-- +countWordReductionsZ2 + :: Int -- ^ g = number of generators in the free group + -> Int -- ^ n = length of the unreduced word + -> Int -- ^ k = length of the reduced word + -> Integer +countWordReductionsZ2 gens_ nn_ kk_ + | nn==0 = if k==0 then 1 else 0 + | even nn && kk == 0 = sum [ ( binomial (nn-i) (n -i) * g^(i ) * (g-1)^(n -i ) * ( i) ) `div` (nn-i) | i<-[0..n ] ] + | even nn && even kk = sum [ ( binomial (nn-i) (n-k-i) * g^(i+1) * (g-1)^(n+k-i-1) * (kk+i) ) `div` (nn-i) | i<-[0..n-k] ] + | odd nn && odd kk = sum [ ( binomial (nn-i) (n-k-i) * g^(i+1) * (g-1)^(n+k-i ) * (kk+i) ) `div` (nn-i) | i<-[0..n-k] ] + | otherwise = 0 + where + g = fromIntegral gens_ :: Integer + nn = fromIntegral nn_ :: Integer + kk = fromIntegral kk_ :: Integer + + n = div nn 2 + k = div kk 2 + +-- | Counts the number of words (without inverse generators) of length @n@ +-- which reduce to the identity element, using the relations @x^3=1@. +-- +-- > countIdentityWordsZ3NoInv g n == sum [ 1 | w <- allWordsNoInv g n, 0 == length (reduceWordZ2 w) ] +-- +-- In mathematica, the formula is: @Sum[ g^k * (g-1)^(n-k) * k/n * Binomial[3*n-k-1, n-k] , {k, 1,n} ]@ +-- +countIdentityWordsZ3NoInv + :: Int -- ^ g = number of generators in the free group + -> Int -- ^ n = length of the unreduced word + -> Integer +countIdentityWordsZ3NoInv gens_ nn_ + | nn==0 = 1 + | mod nn 3 == 0 = sum [ ( binomial (3*n-i-1) (n-i) * g^i * (g-1)^(n-i) * i ) `div` n | i<-[1..n] ] + | otherwise = 0 + where + g = fromIntegral gens_ :: Integer + nn = fromIntegral nn_ :: Integer + + n = div nn 3 + +-------------------------------------------------------------------------------- + +{- + +-- some basic testing. TODO: real tests + +import Math.Combinat.Helper +import Math.Combinat.Groups.Free + +g = 3 :: Int +maxn = 8 :: Int + +bad_free = [ w | n<-[0..maxn] , w <- allWords g n , not (reduceWordFree w `equivalentFree` reduceWordFreeNaive w) ] +bad_z2 = [ w | n<-[0..maxn] , w <- allWords g n , not (reduceWordZ2 w `equivalentZ2` reduceWordZ2Naive w) ] +bad_z3 = [ w | n<-[0..maxn] , w <- allWords g n , not (reduceWordZ3 w `equivalentZ3` reduceWordZ3Naive w) ] +bad_zm m = [ w | n<-[0..maxn] , w <- allWords g n , not (equivalentZm m (reduceWordZm m w) (reduceWordZmNaive m w)) ] + +speed_free = sum' [ length (reduceWordFree w) | n<-[0..maxn] , w <- allWords g n ] +speed_z2 = sum' [ length (reduceWordZ2 w) | n<-[0..maxn] , w <- allWords g n ] +speed_z3 = sum' [ length (reduceWordZ3 w) | n<-[0..maxn] , w <- allWords g n ] +speed_zm m = sum' [ length (reduceWordZm m w) | n<-[0..maxn] , w <- allWords g n ] + +naive_speed_free = sum' [ length (reduceWordFreeNaive w) | n<-[0..maxn] , w <- allWords g n ] +naive_speed_z2 = sum' [ length (reduceWordZ2Naive w) | n<-[0..maxn] , w <- allWords g n ] +naive_speed_z3 = sum' [ length (reduceWordZ3Naive w) | n<-[0..maxn] , w <- allWords g n ] +naive_speed_zm m = sum' [ length (reduceWordZmNaive m w) | n<-[0..maxn] , w <- allWords g n ] + +-} + +-------------------------------------------------------------------------------- + + diff --git a/Math/Combinat/Groups/Thompson/F.hs b/Math/Combinat/Groups/Thompson/F.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Groups/Thompson/F.hs @@ -0,0 +1,404 @@ + +-- | Thompson's group F. +-- +-- See eg. +-- +-- Based mainly on James Michael Belk's PhD thesis \"THOMPSON'S GROUP F\"; +-- see +-- + +{-# LANGUAGE TypeSynonymInstances, FlexibleInstances, BangPatterns, PatternSynonyms, DeriveFunctor #-} +module Math.Combinat.Groups.Thompson.F where + +-------------------------------------------------------------------------------- + +import Data.List + +import Math.Combinat.Classes +import Math.Combinat.ASCII + +import Math.Combinat.Trees.Binary ( BinTree ) +import qualified Math.Combinat.Trees.Binary as B + +-------------------------------------------------------------------------------- +-- * Tree diagrams + +-- | A tree diagram, consisting of two binary trees with the same number of leaves, +-- representing an element of the group F. +data TDiag = TDiag + { _width :: !Int -- ^ the width is the number of leaves, minus 1, of both diagrams + , _domain :: !T -- ^ the top diagram correspond to the /domain/ + , _range :: !T -- ^ the bottom diagram corresponds to the /range/ + } + deriving (Eq,Ord,Show) + +instance DrawASCII TDiag where + ascii = asciiTDiag + +instance HasWidth TDiag where + width = _width + +-- | Creates a tree diagram from two trees +mkTDiag :: T -> T -> TDiag +mkTDiag d1 d2 = reduce $ mkTDiagDontReduce d1 d2 + +-- | Creates a tree diagram, but does not reduce it. +mkTDiagDontReduce :: T -> T -> TDiag +mkTDiagDontReduce top bot = + if w1 == w2 + then TDiag w1 top bot + else error "mkTDiag: widths do not match" + where + w1 = treeWidth top + w2 = treeWidth bot + + +isValidTDiag :: TDiag -> Bool +isValidTDiag (TDiag w top bot) = (treeWidth top == w && treeWidth bot == w) + +isPositive :: TDiag -> Bool +isPositive (TDiag w top bot) = (bot == rightVine w) + +isReduced :: TDiag -> Bool +isReduced diag = (reduce diag == diag) + +-- | The generator x0 +x0 :: TDiag +x0 = TDiag 2 top bot where + top = branch caret leaf + bot = branch leaf caret + +-- | The generator x1 +x1 :: TDiag +x1 = xk 1 + +-- | The generators x0, x1, x2 ... +xk :: Int -> TDiag +xk = go where + go k | k< 0 = error "xk: negative indexed generator" + | k==0 = x0 + | otherwise = let TDiag _ t b = go (k-1) + in TDiag (k+2) (branch leaf t) (branch leaf b) + +-- | The identity element in the group F +identity :: TDiag +identity = TDiag 0 Lf Lf + +-- | A /positive diagram/ is a diagram whose bottom tree (the range) is a right vine. +positive :: T -> TDiag +positive t = TDiag w t (rightVine w) where w = treeWidth t + +-- | Swaps the top and bottom of a tree diagram. This is the inverse in the group F. +-- (Note: we don't do reduction here, as this operation keeps the reducedness) +inverse :: TDiag -> TDiag +inverse (TDiag w top bot) = TDiag w bot top + +-- | Decides whether two (possibly unreduced) tree diagrams represents the same group element in F. +equivalent :: TDiag -> TDiag -> Bool +equivalent diag1 diag2 = (identity == reduce (compose diag1 (inverse diag2))) + +-------------------------------------------------------------------------------- +-- * Reduction of tree diagrams + +-- | Reduces a diagram. The result is a normal form of an element in the group F. +reduce :: TDiag -> TDiag +reduce = worker where + + worker :: TDiag -> TDiag + worker diag = case step diag of + Nothing -> diag + Just diag' -> worker diag' + + step :: TDiag -> Maybe TDiag + step (TDiag w top bot) = + if null idxs + then Nothing + else Just $ TDiag w' top' bot' + where + cs1 = treeCaretList top + cs2 = treeCaretList bot + idxs = sortedIntersect cs1 cs2 + w' = w - length idxs + top' = removeCarets idxs top + bot' = removeCarets idxs bot + + -- | Intersects sorted lists + sortedIntersect :: [Int] -> [Int] -> [Int] + sortedIntersect = go where + go [] _ = [] + go _ [] = [] + go xxs@(x:xs) yys@(y:ys) = case compare x y of + LT -> go xs yys + EQ -> x : go xs ys + GT -> go xxs ys + +-- | List of carets at the bottom of the tree, indexed by their left edge position +treeCaretList :: T -> [Int] +treeCaretList = snd . go 0 where + go !x t = case t of + Lf -> (x+1 , [] ) + Ct -> (x+2 , [x] ) + Br t1 t2 -> (x2 , cs1++cs2) where + (x1 , cs1) = go x t1 + (x2 , cs2) = go x1 t2 + +-- | Remove the carets with the given indices +-- (throws an error if there is no caret at the given index) +removeCarets :: [Int] -> T -> T +removeCarets idxs tree = if null rem then final else error ("removeCarets: some stuff remained: " ++ show rem) where + + (_,rem,final) = go 0 idxs tree where + + go :: Int -> [Int] -> T -> (Int,[Int],T) + go !x [] t = (x + treeWidth t , [] , t) + go !x iis@(i:is) t = case t of + Lf -> (x+1 , iis , t) + Ct -> if x==i then (x+2 , is , Lf) else (x+2 , iis , Ct) + Br t1 t2 -> (x2 , iis2 , Br t1' t2') where + (x1 , iis1 , t1') = go x iis t1 + (x2 , iis2 , t2') = go x1 iis1 t2 + +-------------------------------------------------------------------------------- +-- * Composition of tree diagrams + +-- | If @diag1@ corresponds to the PL function @f@, and @diag2@ to @g@, then @compose diag1 diag2@ +-- will correspond to @(g.f)@ (note that the order is opposite than normal function composition!) +-- +-- This is the multiplication in the group F. +-- +compose :: TDiag -> TDiag -> TDiag +compose d1 d2 = reduce (composeDontReduce d1 d2) + +-- | Compose two tree diagrams without reducing the result +composeDontReduce :: TDiag -> TDiag -> TDiag +composeDontReduce (TDiag w1 top1 bot1) (TDiag w2 top2 bot2) = new where + new = mkTDiagDontReduce top' bot' + (list1,list2) = extensionToCommonTree bot1 top2 + top' = listGraft list1 top1 + bot' = listGraft list2 bot2 + +-- | Given two binary trees, we return a pair of list of subtrees which, grafted the to leaves of +-- the first (resp. the second) tree, results in the same extended tree. +extensionToCommonTree :: T -> T -> ([T],[T]) +extensionToCommonTree t1 t2 = snd $ go (0,0) (t1,t2) where + go (!x1,!x2) (!t1,!t2) = + case (t1,t2) of + ( Lf , Lf ) -> ( (x1+n1 , x2+n2 ) , ( [Lf] , [Lf] ) ) + ( Lf , Br _ _ ) -> ( (x1+n1 , x2+n2 ) , ( [t2] , replicate n2 Lf ) ) + ( Br _ _ , Lf ) -> ( (x1+n1 , x2+n2 ) , ( replicate n1 Lf , [t1] ) ) + ( Br l1 r1 , Br l2 r2 ) + -> let ( (x1' ,x2' ) , (ps1,ps2) ) = go (x1 ,x2 ) (l1,l2) + ( (x1'',x2'') , (qs1,qs2) ) = go (x1',x2') (r1,r2) + in ( (x1'',x2'') , (ps1++qs1, ps2++qs2) ) + where + n1 = numberOfLeaves t1 + n2 = numberOfLeaves t2 + +-------------------------------------------------------------------------------- +-- * Subdivions + +-- | Returns the list of dyadic subdivision points +subdivision1 :: T -> [Rational] +subdivision1 = go 0 1 where + go !a !b t = case t of + Leaf _ -> [a,b] + Branch l r -> go a c l ++ tail (go c b r) where c = (a+b)/2 + +-- | Returns the list of dyadic intervals +subdivision2 :: T -> [(Rational,Rational)] +subdivision2 = go 0 1 where + go !a !b t = case t of + Leaf _ -> [(a,b)] + Branch l r -> go a c l ++ go c b r where c = (a+b)/2 + + +-------------------------------------------------------------------------------- +-- * Binary trees + +-- | A (strict) binary tree with labelled leaves (but unlabelled nodes) +data Tree a + = Branch !(Tree a) !(Tree a) + | Leaf !a + deriving (Eq,Ord,Show,Functor) + +-- | The monadic join operation of binary trees +graft :: Tree (Tree a) -> Tree a +graft = go where + go (Branch l r) = Branch (go l) (go r) + go (Leaf t ) = t + +-- | A list version of 'graft' +listGraft :: [Tree a] -> Tree b -> Tree a +listGraft subs big = snd $ go subs big where + go ggs@(g:gs) t = case t of + Leaf _ -> (gs,g) + Branch l r -> (gs2, Branch l' r') where + (gs1,l') = go ggs l + (gs2,r') = go gs1 r + +-- | A completely unlabelled binary tree +type T = Tree () + +instance DrawASCII T where + ascii = asciiT + +instance HasNumberOfLeaves (Tree a) where + numberOfLeaves = treeNumberOfLeaves + +instance HasWidth (Tree a) where + width = treeWidth + +leaf :: T +leaf = Leaf () + +branch :: T -> T -> T +branch = Branch + +caret :: T +caret = branch leaf leaf + +treeNumberOfLeaves :: Tree a -> Int +treeNumberOfLeaves = go where + go (Branch l r) = go l + go r + go (Leaf _ ) = 1 + +-- | The width of the tree is the number of leaves minus 1. +treeWidth :: Tree a -> Int +treeWidth t = numberOfLeaves t - 1 + +-- | Enumerates the leaves a tree, starting from 0 +enumerate_ :: Tree a -> Tree Int +enumerate_ = snd . enumerate + +-- | Enumerates the leaves a tree, and also returns the number of leaves +enumerate :: Tree a -> (Int, Tree Int) +enumerate = go 0 where + go !k t = case t of + Leaf _ -> (k+1 , Leaf k) + Branch l r -> let (k' ,l') = go k l + (k'',r') = go k' r + in (k'', Branch l' r') + +-- | \"Right vine\" of the given width +rightVine :: Int -> T +rightVine k + | k< 0 = error "rightVine: negative width" + | k==0 = leaf + | otherwise = branch leaf (rightVine (k-1)) + +-- | \"Left vine\" of the given width +leftVine :: Int -> T +leftVine k + | k< 0 = error "leftVine: negative width" + | k==0 = leaf + | otherwise = branch (leftVine (k-1)) leaf + +-- | Flips each node of a binary tree +flipTree :: Tree a -> Tree a +flipTree = go where + go t = case t of + Leaf _ -> t + Branch l r -> Branch (go r) (go l) + +-------------------------------------------------------------------------------- +-- * Conversion to\/from BinTree + +-- | 'Tree' and 'BinTree' are the same type, except that 'Tree' is strict. +-- +-- TODO: maybe unify these two types? Until that, you can convert between the two +-- with these functions if necessary. +-- +toBinTree :: Tree a -> B.BinTree a +toBinTree = go where + go (Branch l r) = B.Branch (go l) (go r) + go (Leaf y ) = B.Leaf y + +fromBinTree :: B.BinTree a -> Tree a +fromBinTree = go where + go (B.Branch l r) = Branch (go l) (go r) + go (B.Leaf y ) = Leaf y + +-------------------------------------------------------------------------------- +-- * Pattern synonyms + +pattern Lf = Leaf () +pattern Br l r = Branch l r +pattern Ct = Br Lf Lf +pattern X0 = TDiag 2 (Br Ct Lf) (Br Lf Ct) +pattern X1 = TDiag 3 (Br Lf (Br Ct Lf)) (Br Lf (Br Lf Ct)) + +-------------------------------------------------------------------------------- +-- * ASCII + +-- | Draws a binary tree, with all leaves at the same (bottom) row +asciiT :: T -> ASCII +asciiT = asciiT' False + +-- | Draws a binary tree; when the boolean flag is @True@, we draw upside down +asciiT' :: Bool -> T -> ASCII +asciiT' inv = go where + + go t = case t of + Leaf _ -> emptyRect + Branch l r -> + if yl >= yr + then pasteOnto (yl+yr+1,if inv then yr else 0) (rs $ yl+1) $ + vcat HCenter + (bc $ yr+1) + (hcat bot al ar) + else pasteOnto (yl, if inv then yl else 0) (ls $ yr+1) $ + vcat HCenter + (bc $ yl+1) + (hcat bot al ar) + where + al = go l + ar = go r + yl = asciiYSize al + yr = asciiYSize ar + + bot = if inv then VTop else VBottom + hcat align p q = hCatWith align (HSepString " ") [p,q] + vcat align p q = vCatWith align VSepEmpty $ if inv then [q,p] else [p,q] + bc = if inv then asciiBigInvCaret else asciiBigCaret + ls = if inv then asciiBigRightSlope else asciiBigLeftSlope + rs = if inv then asciiBigLeftSlope else asciiBigRightSlope + + asciiBigCaret :: Int -> ASCII + asciiBigCaret k = hCatWith VTop HSepEmpty [ asciiBigLeftSlope k , asciiBigRightSlope k ] + + asciiBigInvCaret :: Int -> ASCII + asciiBigInvCaret k = hCatWith VTop HSepEmpty [ asciiBigRightSlope k , asciiBigLeftSlope k ] + + asciiBigLeftSlope :: Int -> ASCII + asciiBigLeftSlope k = if k>0 + then asciiFromLines [ replicate l ' ' ++ "/" | l<-[k-1,k-2..0] ] + else emptyRect + + asciiBigRightSlope :: Int -> ASCII + asciiBigRightSlope k = if k>0 + then asciiFromLines [ replicate l ' ' ++ "\\" | l<-[0..k-1] ] + else emptyRect + +-- | Draws a binary tree, with all leaves at the same (bottom) row, and labelling +-- the leaves starting with 0 (continuing with letters after 9) +asciiTLabels :: T -> ASCII +asciiTLabels = asciiTLabels' False + +-- | When the flag is true, we draw upside down +asciiTLabels' :: Bool -> T -> ASCII +asciiTLabels' inv t = + if inv + then vCatWith HLeft VSepEmpty [ labels , asciiT' inv t ] + else vCatWith HLeft VSepEmpty [ asciiT' inv t , labels ] + where + w = treeWidth t + labels = asciiFromString $ intersperse ' ' $ take (w+1) allLabels + allLabels = ['0'..'9'] ++ ['a'..'z'] + +-- | Draws a tree diagram +asciiTDiag :: TDiag -> ASCII +asciiTDiag (TDiag _ top bot) = vCatWith HLeft (VSepString " ") [asciiT' False top , asciiT' True bot] + +-------------------------------------------------------------------------------- + + diff --git a/Math/Combinat/Helper.hs b/Math/Combinat/Helper.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Helper.hs @@ -0,0 +1,280 @@ + +-- | Miscellaneous helper functions + +{-# LANGUAGE BangPatterns, PolyKinds, GeneralizedNewtypeDeriving #-} +module Math.Combinat.Helper where + +-------------------------------------------------------------------------------- + +import Control.Monad +import Control.Applicative ( Applicative(..) ) -- required before AMP (before GHC 7.10) +import Data.Functor.Identity + +import Data.List +import Data.Ord +import Data.Proxy + +import Data.Set (Set) ; import qualified Data.Set as Set +import Data.Map (Map) ; import qualified Data.Map as Map + +import Debug.Trace + +import System.Random +import Control.Monad.Trans.State + +-------------------------------------------------------------------------------- +-- * debugging + +debug :: Show a => a -> b -> b +debug x y = trace ("-- " ++ show x ++ "\n") y + +-------------------------------------------------------------------------------- +-- * pairs + +swap :: (a,b) -> (b,a) +swap (x,y) = (y,x) + +pairs :: [a] -> [(a,a)] +pairs = go where + go (x:xs@(y:_)) = (x,y) : go xs + go _ = [] + +pairsWith :: (a -> a -> b) -> [a] -> [b] +pairsWith f = go where + go (x:xs@(y:_)) = f x y : go xs + go _ = [] + +-------------------------------------------------------------------------------- +-- * lists + +{-# SPECIALIZE sum' :: [Int] -> Int #-} +{-# SPECIALIZE sum' :: [Integer] -> Integer #-} +sum' :: Num a => [a] -> a +sum' = foldl' (+) 0 + +-------------------------------------------------------------------------------- +-- * equality and ordering + +equating :: Eq b => (a -> b) -> a -> a -> Bool +equating f x y = (f x == f y) + +reverseOrdering :: Ordering -> Ordering +reverseOrdering LT = GT +reverseOrdering GT = LT +reverseOrdering EQ = EQ + +reverseCompare :: Ord a => a -> a -> Ordering +reverseCompare x y = reverseOrdering $ compare x y + +reverseSort :: Ord a => [a] -> [a] +reverseSort = sortBy reverseCompare + +groupSortBy :: (Eq b, Ord b) => (a -> b) -> [a] -> [[a]] +groupSortBy f = groupBy (equating f) . sortBy (comparing f) + +nubOrd :: Ord a => [a] -> [a] +nubOrd = worker Set.empty where + worker _ [] = [] + worker s (x:xs) + | Set.member x s = worker s xs + | otherwise = x : worker (Set.insert x s) xs + +-------------------------------------------------------------------------------- +-- * increasing \/ decreasing sequences + +{-# SPECIALIZE isWeaklyIncreasing :: [Int] -> Bool #-} +isWeaklyIncreasing :: Ord a => [a] -> Bool +isWeaklyIncreasing = go where + go xs = case xs of + (a:rest@(b:_)) -> a <= b && go rest + [_] -> True + [] -> True + +{-# SPECIALIZE isStrictlyIncreasing :: [Int] -> Bool #-} +isStrictlyIncreasing :: Ord a => [a] -> Bool +isStrictlyIncreasing = go where + go xs = case xs of + (a:rest@(b:_)) -> a < b && go rest + [_] -> True + [] -> True + +{-# SPECIALIZE isWeaklyDecreasing :: [Int] -> Bool #-} +isWeaklyDecreasing :: Ord a => [a] -> Bool +isWeaklyDecreasing = go where + go xs = case xs of + (a:rest@(b:_)) -> a >= b && go rest + [_] -> True + [] -> True + +{-# SPECIALIZE isStrictlyDecreasing :: [Int] -> Bool #-} +isStrictlyDecreasing :: Ord a => [a] -> Bool +isStrictlyDecreasing = go where + go xs = case xs of + (a:rest@(b:_)) -> a > b && go rest + [_] -> True + [] -> True + +-------------------------------------------------------------------------------- +-- * first \/ last + +-- | The boolean argument will @True@ only for the last element +mapWithLast :: (Bool -> a -> b) -> [a] -> [b] +mapWithLast f = go where + go (x : []) = f True x : [] + go (x : xs) = f False x : go xs + +mapWithFirst :: (Bool -> a -> b) -> [a] -> [b] +mapWithFirst f = go True where + go b (x:xs) = f b x : go False xs + +mapWithFirstLast :: (Bool -> Bool -> a -> b) -> [a] -> [b] +mapWithFirstLast f = go True where + go b (x : []) = f b True x : [] + go b (x : xs) = f b False x : go False xs + +-------------------------------------------------------------------------------- +-- * older helpers for ASCII drawing + +-- | extend lines with spaces so that they have the same line +mkLinesUniformWidth :: [String] -> [String] +mkLinesUniformWidth old = zipWith worker ls old where + ls = map length old + m = maximum ls + worker l s = s ++ replicate (m-l) ' ' + +mkBlocksUniformHeight :: [[String]] -> [[String]] +mkBlocksUniformHeight old = zipWith worker ls old where + ls = map length old + m = maximum ls + worker l s = s ++ replicate (m-l) "" + +mkUniformBlocks :: [[String]] -> [[String]] +mkUniformBlocks = map mkLinesUniformWidth . mkBlocksUniformHeight + +hConcatLines :: [[String]] -> [String] +hConcatLines = map concat . transpose . mkUniformBlocks + +vConcatLines :: [[String]] -> [String] +vConcatLines = concat + +-------------------------------------------------------------------------------- +-- * counting + +-- helps testing the random rutines +count :: Eq a => a -> [a] -> Int +count x xs = length $ filter (==x) xs + +histogram :: (Eq a, Ord a) => [a] -> [(a,Int)] +histogram xs = Map.toList table where + table = Map.fromListWith (+) [ (x,1) | x<-xs ] + +-------------------------------------------------------------------------------- +-- * maybe + +fromJust :: Maybe a -> a +fromJust (Just x) = x +fromJust Nothing = error "fromJust: Nothing" + +-------------------------------------------------------------------------------- +-- * bool + +intToBool :: Int -> Bool +intToBool 0 = False +intToBool 1 = True +intToBool _ = error "intToBool" + +boolToInt :: Bool -> Int +boolToInt False = 0 +boolToInt True = 1 + +-------------------------------------------------------------------------------- +-- * iteration + +-- iterated function application +nest :: Int -> (a -> a) -> a -> a +nest !0 _ x = x +nest !n f x = nest (n-1) f (f x) + +unfold1 :: (a -> Maybe a) -> a -> [a] +unfold1 f x = case f x of + Nothing -> [x] + Just y -> x : unfold1 f y + +unfold :: (b -> (a,Maybe b)) -> b -> [a] +unfold f y = let (x,m) = f y in case m of + Nothing -> [x] + Just y' -> x : unfold f y' + +unfoldEither :: (b -> Either c (b,a)) -> b -> (c,[a]) +unfoldEither f y = case f y of + Left z -> (z,[]) + Right (y,x) -> let (z,xs) = unfoldEither f y in (z,x:xs) + +unfoldM :: Monad m => (b -> m (a,Maybe b)) -> b -> m [a] +unfoldM f y = do + (x,m) <- f y + case m of + Nothing -> return [x] + Just y' -> do + xs <- unfoldM f y' + return (x:xs) + +mapAccumM :: Monad m => (acc -> x -> m (acc, y)) -> acc -> [x] -> m (acc, [y]) +mapAccumM _ s [] = return (s, []) +mapAccumM f s (x:xs) = do + (s1,y) <- f s x + (s2,ys) <- mapAccumM f s1 xs + return (s2, y:ys) + +-------------------------------------------------------------------------------- +-- * long zipwith + +longZipWith :: a -> b -> (a -> b -> c) -> [a] -> [b] -> [c] +longZipWith a0 b0 f = go where + go (x:xs) (y:ys) = f x y : go xs ys + go [] ys = [ f a0 y | y<-ys ] + go xs [] = [ f x b0 | x<-xs ] + +{- +longZipWithZero :: (Num a, Num b) => (a -> b -> c) -> [a] -> [b] -> [c] +longZipWithZero = longZipWith 0 0 +-} + +-------------------------------------------------------------------------------- +-- * random + +-- | A simple random monad to make life suck less +type Rand g = RandT g Identity + +runRand :: Rand g a -> g -> (a,g) +runRand action g = runIdentity (runRandT action g) + +flipRunRand :: Rand s a -> s -> (s,a) +flipRunRand action g = runIdentity (flipRunRandT action g) + + +-- | The Rand monad transformer +newtype RandT g m a = RandT (StateT g m a) deriving (Functor,Applicative,Monad) + +runRandT :: RandT g m a -> g -> m (a,g) +runRandT (RandT stuff) = runStateT stuff + +-- | This may be occasionally useful +flipRunRandT :: Monad m => RandT s m a -> s -> m (s,a) +flipRunRandT action ini = liftM swap $ runRandT action ini + + +-- | Puts a standard-conforming random function into the monad +rand :: (g -> (a,g)) -> Rand g a +rand user = RandT (state user) + +randRoll :: (RandomGen g, Random a) => Rand g a +randRoll = rand random + +randChoose :: (RandomGen g, Random a) => (a,a) -> Rand g a +randChoose uv = rand (randomR uv) + +randProxy1 :: Rand g (f n) -> Proxy n -> Rand g (f n) +randProxy1 action _ = action + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/LatticePaths.hs b/Math/Combinat/LatticePaths.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/LatticePaths.hs @@ -0,0 +1,386 @@ + +-- | Dyck paths, lattice paths, etc +-- +-- For example, the following figure represents a Dyck path of height 5 with 3 zero-touches (not counting the starting point, +-- but counting the endpoint) and 7 peaks: +-- +-- <> +-- + +{-# LANGUAGE BangPatterns, FlexibleInstances, TypeSynonymInstances #-} +module Math.Combinat.LatticePaths where + +-------------------------------------------------------------------------------- + +import Data.List +import System.Random + +import Math.Combinat.Classes +import Math.Combinat.Numbers +import Math.Combinat.Trees.Binary +import Math.Combinat.ASCII as ASCII + +-------------------------------------------------------------------------------- +-- * Types + +-- | A step in a lattice path +data Step + = UpStep -- ^ the step @(1,1)@ + | DownStep -- ^ the step @(1,-1)@ + deriving (Eq,Ord,Show) + +-- | A lattice path is a path using only the allowed steps, never going below the zero level line @y=0@. +-- +-- Note that if you rotate such a path by 45 degrees counterclockwise, +-- you get a path which uses only the steps @(1,0)@ and @(0,1)@, and stays +-- above the main diagonal (hence the name, we just use a different convention). +-- +type LatticePath = [Step] + +-------------------------------------------------------------------------------- +-- * ascii drawing of paths + +-- | Draws the path into a list of lines. For example try: +-- +-- > autotabulate RowMajor (Right 5) (map asciiPath $ dyckPaths 4) +-- +asciiPath :: LatticePath -> ASCII +asciiPath p = asciiFromLines $ transpose (go 0 p) where + + go !h [] = [] + go !h (x:xs) = case x of + UpStep -> ee h x : go (h+1) xs + DownStep -> ee (h-1) x : go (h-1) xs + + maxh = pathHeight p + + ee h x = replicate (maxh-h-1) ' ' ++ [ch x] ++ replicate h ' ' + ch x = case x of + UpStep -> '/' + DownStep -> '\\' + +instance DrawASCII LatticePath where + ascii = asciiPath + +-------------------------------------------------------------------------------- +-- * elementary queries + +-- | A lattice path is called \"valid\", if it never goes below the @y=0@ line. +isValidPath :: LatticePath -> Bool +isValidPath = go 0 where + go :: Int -> LatticePath -> Bool + go !y [] = y>=0 + go !y (t:ts) = let y' = case t of { UpStep -> y+1 ; DownStep -> y-1 } + in if y'<0 then False + else go y' ts + +-- | A Dyck path is a lattice path whose last point lies on the @y=0@ line +isDyckPath :: LatticePath -> Bool +isDyckPath = go 0 where + go :: Int -> LatticePath -> Bool + go !y [] = y==0 + go !y (t:ts) = let y' = case t of { UpStep -> y+1 ; DownStep -> y-1 } + in if y'<0 then False + else go y' ts + +-- | Maximal height of a lattice path +pathHeight :: LatticePath -> Int +pathHeight = go 0 0 where + go :: Int -> Int -> LatticePath -> Int + go !h !y [] = h + go !h !y (t:ts) = case t of + UpStep -> go (max h (y+1)) (y+1) ts + DownStep -> go h (y-1) ts + +instance HasHeight LatticePath where + height = pathHeight + +instance HasWidth LatticePath where + width = length + +-- | Endpoint of a lattice path, which starts from @(0,0)@. +pathEndpoint :: LatticePath -> (Int,Int) +pathEndpoint = go 0 0 where + go :: Int -> Int -> LatticePath -> (Int,Int) + go !x !y [] = (x,y) + go !x !y (t:ts) = case t of + UpStep -> go (x+1) (y+1) ts + DownStep -> go (x+1) (y-1) ts + +-- | Returns the coordinates of the path (excluding the starting point @(0,0)@, but including +-- the endpoint) +pathCoordinates :: LatticePath -> [(Int,Int)] +pathCoordinates = go 0 0 where + go :: Int -> Int -> LatticePath -> [(Int,Int)] + go _ _ [] = [] + go !x !y (t:ts) = let x' = x + 1 + y' = case t of { UpStep -> y+1 ; DownStep -> y-1 } + in (x',y') : go x' y' ts + +-- | Counts the up-steps +pathNumberOfUpSteps :: LatticePath -> Int +pathNumberOfUpSteps = fst . pathNumberOfUpDownSteps + +-- | Counts the down-steps +pathNumberOfDownSteps :: LatticePath -> Int +pathNumberOfDownSteps = snd . pathNumberOfUpDownSteps + +-- | Counts both the up-steps and down-steps +pathNumberOfUpDownSteps :: LatticePath -> (Int,Int) +pathNumberOfUpDownSteps = go 0 0 where + go :: Int -> Int -> LatticePath -> (Int,Int) + go !u !d (p:ps) = case p of + UpStep -> go (u+1) d ps + DownStep -> go u (d+1) ps + go !u !d [] = (u,d) + +-------------------------------------------------------------------------------- +-- * path-specific queries + +-- | Number of peaks of a path (excluding the endpoint) +pathNumberOfPeaks :: LatticePath -> Int +pathNumberOfPeaks = go 0 where + go :: Int -> LatticePath -> Int + go !k (x:xs@(y:_)) = go (if x==UpStep && y==DownStep then k+1 else k) xs + go !k [x] = k + go !k [ ] = k + +-- | Number of points on the path which touch the @y=0@ zero level line +-- (excluding the starting point @(0,0)@, but including the endpoint; that is, for Dyck paths it this is always positive!). +pathNumberOfZeroTouches :: LatticePath -> Int +pathNumberOfZeroTouches = pathNumberOfTouches' 0 + +-- | Number of points on the path which touch the level line at height @h@ +-- (excluding the starting point @(0,0)@, but including the endpoint). +pathNumberOfTouches' + :: Int -- ^ @h@ = the touch level + -> LatticePath -> Int +pathNumberOfTouches' h = go 0 0 0 where + go :: Int -> Int -> Int -> LatticePath -> Int + go !cnt _ _ [] = cnt + go !cnt !x !y (t:ts) = let y' = case t of { UpStep -> y+1 ; DownStep -> y-1 } + cnt' = if y'==h then cnt+1 else cnt + in go cnt' (x+1) y' ts + +-------------------------------------------------------------------------------- +-- * Dyck paths + +-- | @dyckPaths m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@. +-- +-- Remark: Dyck paths are obviously in bijection with nested parentheses, and thus +-- also with binary trees. +-- +-- Order is reverse lexicographical: +-- +-- > sort (dyckPaths m) == reverse (dyckPaths m) +-- +dyckPaths :: Int -> [LatticePath] +dyckPaths = map nestedParensToDyckPath . nestedParentheses + +-- | @dyckPaths m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@. +-- +-- > sort (dyckPathsNaive m) == sort (dyckPaths m) +-- +-- Naive recursive algorithm, order is ad-hoc +-- +dyckPathsNaive :: Int -> [LatticePath] +dyckPathsNaive = worker where + worker 0 = [[]] + worker m = as ++ bs where + as = [ bracket p | p <- worker (m-1) ] + bs = [ bracket p ++ q | k <- [1..m-1] , p <- worker (k-1) , q <- worker (m-k) ] + bracket p = UpStep : p ++ [DownStep] + +-- | The number of Dyck paths from @(0,0)@ to @(2m,0)@ is simply the m\'th Catalan number. +countDyckPaths :: Int -> Integer +countDyckPaths m = catalan m + +-- | The trivial bijection +nestedParensToDyckPath :: [Paren] -> LatticePath +nestedParensToDyckPath = map f where + f p = case p of { LeftParen -> UpStep ; RightParen -> DownStep } + +-- | The trivial bijection in the other direction +dyckPathToNestedParens :: LatticePath -> [Paren] +dyckPathToNestedParens = map g where + g s = case s of { UpStep -> LeftParen ; DownStep -> RightParen } + +-------------------------------------------------------------------------------- +-- * Bounded Dyck paths + +-- | @boundedDyckPaths h m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@ whose height is at most @h@. +-- Synonym for 'boundedDyckPathsNaive'. +-- +boundedDyckPaths + :: Int -- ^ @h@ = maximum height + -> Int -- ^ @m@ = half-length + -> [LatticePath] +boundedDyckPaths = boundedDyckPathsNaive + +-- | @boundedDyckPathsNaive h m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@ whose height is at most @h@. +-- +-- > sort (boundedDyckPaths h m) == sort [ p | p <- dyckPaths m , pathHeight p <= h ] +-- > sort (boundedDyckPaths m m) == sort (dyckPaths m) +-- +-- Naive recursive algorithm, resulting order is pretty ad-hoc. +-- +boundedDyckPathsNaive + :: Int -- ^ @h@ = maximum height + -> Int -- ^ @m@ = half-length + -> [LatticePath] +boundedDyckPathsNaive = worker where + worker !h !m + | h<0 = [] + | m<0 = [] + | m==0 = [[]] + | h<=0 = [] + | otherwise = as ++ bs + where + bracket p = UpStep : p ++ [DownStep] + as = [ bracket p | p <- boundedDyckPaths (h-1) (m-1) ] + bs = [ bracket p ++ q | k <- [1..m-1] , p <- boundedDyckPaths (h-1) (k-1) , q <- boundedDyckPaths h (m-k) ] + +-------------------------------------------------------------------------------- +-- * More general lattice paths + +-- | All lattice paths from @(0,0)@ to @(x,y)@. Clearly empty unless @x-y@ is even. +-- Synonym for 'latticePathsNaive' +-- +latticePaths :: (Int,Int) -> [LatticePath] +latticePaths = latticePathsNaive + +-- | All lattice paths from @(0,0)@ to @(x,y)@. Clearly empty unless @x-y@ is even. +-- +-- Note that +-- +-- > sort (dyckPaths n) == sort (latticePaths (0,2*n)) +-- +-- Naive recursive algorithm, resulting order is pretty ad-hoc. +-- +latticePathsNaive :: (Int,Int) -> [LatticePath] +latticePathsNaive (x,y) = worker x y where + worker !x !y + | odd (x-y) = [] + | x<0 = [] + | y<0 = [] + | y==0 = dyckPaths (div x 2) + | x==1 && y==1 = [[UpStep]] + | otherwise = as ++ bs + where + bracket p = UpStep : p ++ [DownStep] + as = [ UpStep : p | p <- worker (x-1) (y-1) ] + bs = [ bracket p ++ q | k <- [1..(div x 2)] , p <- dyckPaths (k-1) , q <- worker (x-2*k) y ] + +-- | Lattice paths are counted by the numbers in the Catalan triangle. +countLatticePaths :: (Int,Int) -> Integer +countLatticePaths (x,y) + | even (x+y) = catalanTriangle (div (x+y) 2) (div (x-y) 2) + | otherwise = 0 + +-------------------------------------------------------------------------------- +-- * Zero-level touches + +-- | @touchingDyckPaths k m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@ which touch the +-- zero level line @y=0@ exactly @k@ times (excluding the starting point, but including the endpoint; +-- thus, @k@ should be positive). Synonym for 'touchingDyckPathsNaive'. +touchingDyckPaths + :: Int -- ^ @k@ = number of zero-touches + -> Int -- ^ @m@ = half-length + -> [LatticePath] +touchingDyckPaths = touchingDyckPathsNaive + + +-- | @touchingDyckPathsNaive k m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@ which touch the +-- zero level line @y=0@ exactly @k@ times (excluding the starting point, but including the endpoint; +-- thus, @k@ should be positive). +-- +-- > sort (touchingDyckPathsNaive k m) == sort [ p | p <- dyckPaths m , pathNumberOfZeroTouches p == k ] +-- +-- Naive recursive algorithm, resulting order is pretty ad-hoc. +-- +touchingDyckPathsNaive + :: Int -- ^ @k@ = number of zero-touches + -> Int -- ^ @m@ = half-length + -> [LatticePath] +touchingDyckPathsNaive = worker where + worker !k !m + | m == 0 = if k==0 then [[]] else [] + | k <= 0 = [] + | m < 0 = [] + | k == 1 = [ bracket p | p <- dyckPaths (m-1) ] + | otherwise = [ bracket p ++ q | l <- [1..m-1] , p <- dyckPaths (l-1) , q <- worker (k-1) (m-l) ] + where + bracket p = UpStep : p ++ [DownStep] + + +-- | There is a bijection from the set of non-empty Dyck paths of length @2n@ which touch the zero lines @t@ times, +-- to lattice paths from @(0,0)@ to @(2n-t-1,t-1)@ (just remove all the down-steps just before touching +-- the zero line, and also the very first up-step). This gives us a counting formula. +countTouchingDyckPaths + :: Int -- ^ @k@ = number of zero-touches + -> Int -- ^ @m@ = half-length + -> Integer +countTouchingDyckPaths t n + | t==0 && n==0 = 1 + | otherwise = countLatticePaths (2*n-t-1,t-1) + +-------------------------------------------------------------------------------- +-- * Dyck paths with given number of peaks + +-- | @peakingDyckPaths k m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@ with exactly @k@ peaks. +-- +-- Synonym for 'peakingDyckPathsNaive' +-- +peakingDyckPaths + :: Int -- ^ @k@ = number of peaks + -> Int -- ^ @m@ = half-length + -> [LatticePath] +peakingDyckPaths = peakingDyckPathsNaive + +-- | @peakingDyckPathsNaive k m@ lists all Dyck paths from @(0,0)@ to @(2m,0)@ with exactly @k@ peaks. +-- +-- > sort (peakingDyckPathsNaive k m) = sort [ p | p <- dyckPaths m , pathNumberOfPeaks p == k ] +-- +-- Naive recursive algorithm, resulting order is pretty ad-hoc. +-- +peakingDyckPathsNaive + :: Int -- ^ @k@ = number of peaks + -> Int -- ^ @m@ = half-length + -> [LatticePath] +peakingDyckPathsNaive = worker where + worker !k !m + | m == 0 = if k==0 then [[]] else [] + | k <= 0 = [] + | m < 0 = [] + | k == 1 = [ singlePeak m ] + | otherwise = as ++ bs ++ cs + where + as = [ bracket p | p <- worker k (m-1) ] + bs = [ smallHill ++ q | q <- worker (k-1) (m-1) ] + cs = [ bracket p ++ q | l <- [2..m-1] , a <- [1..k-1] , p <- worker a (l-1) , q <- worker (k-a) (m-l) ] + smallHill = [ UpStep , DownStep ] + singlePeak !m = replicate m UpStep ++ replicate m DownStep + bracket p = UpStep : p ++ [DownStep] + +-- | Dyck paths of length @2m@ with @k@ peaks are counted by the Narayana numbers @N(m,k) = \binom{m}{k} \binom{m}{k-1} / m@ +countPeakingDyckPaths + :: Int -- ^ @k@ = number of peaks + -> Int -- ^ @m@ = half-length + -> Integer +countPeakingDyckPaths k m + | m == 0 = if k==0 then 1 else 0 + | k <= 0 = 0 + | m < 0 = 0 + | k == 1 = 1 + | otherwise = div (binomial m k * binomial m (k-1)) (fromIntegral m) + +-------------------------------------------------------------------------------- +-- * Random lattice paths + +-- | A uniformly random Dyck path of length @2m@ +randomDyckPath :: RandomGen g => Int -> g -> (LatticePath,g) +randomDyckPath m g0 = (nestedParensToDyckPath parens, g1) where + (parens,g1) = randomNestedParentheses m g0 + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Numbers.hs b/Math/Combinat/Numbers.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Numbers.hs @@ -0,0 +1,194 @@ + +-- | A few important number sequences. +-- +-- See the \"On-Line Encyclopedia of Integer Sequences\", +-- . + +module Math.Combinat.Numbers where + +-------------------------------------------------------------------------------- + +import Data.Array + +import Math.Combinat.Helper ( sum' ) +import Math.Combinat.Sign + +-------------------------------------------------------------------------------- + +-- | A000142. +factorial :: Integral a => a -> Integer +factorial n + | n < 0 = error "factorial: input should be nonnegative" + | n == 0 = 1 + | otherwise = product [1..fromIntegral n] + +-- | A006882. +doubleFactorial :: Integral a => a -> Integer +doubleFactorial n + | n < 0 = error "doubleFactorial: input should be nonnegative" + | n == 0 = 1 + | odd n = product [1,3..fromIntegral n] + | otherwise = product [2,4..fromIntegral n] + +-- | A007318. Note: This is zero for @n<0@ or @k<0@; see also 'signedBinomial' below. +binomial :: Integral a => a -> a -> Integer +binomial n k + | k > n = 0 + | k < 0 = 0 + | k > (n `div` 2) = binomial n (n-k) + | otherwise = (product [n'-k'+1 .. n']) `div` (product [1..k']) + where + k' = fromIntegral k + n' = fromIntegral n + +-- | The extension of the binomial function to negative inputs. This should satisfy the following properties: +-- +-- > for n,k >=0 : signedBinomial n k == binomial n k +-- > for any n,k : signedBinomial n k == signedBinomial n (n-k) +-- > for k >= 0 : signedBinomial (-n) k == (-1)^k * signedBinomial (n+k-1) k +-- +-- Note: This is compatible with Mathematica's @Binomial@ function. +-- +signedBinomial :: Int -> Int -> Integer +signedBinomial n k + | n >= 0 = binomial n k + | k >= 0 = negateIfOdd k $ binomial (k-n-1) k + | otherwise = negateIfOdd (n+k) $ binomial (-k-1) (-n-1) + +{- +test_signed_0 = [ signedBinomial ( n) k == signedBinomial ( n) ( n-k) | n<-[-30..40] , k<-[-30..40] ] +test_signed_1 = [ signedBinomial (-n) k == signedBinomial (-n) (-n-k) | n<-[-30..40] , k<-[-30..40] ] +test_signed_2 = [ signedBinomial (-n) k == negateIfOdd k $ signedBinomial (n+k-1) k | n<-[-30..40] , k<-[0..30] ] +-} + +-- | A given row of the Pascal triangle; equivalent to a sequence of binomial +-- numbers, but much more efficient. You can also left-fold over it. +-- +-- > pascalRow n == [ binomial n k | k<-[0..n] ] +pascalRow :: Integral a => a -> [Integer] +pascalRow n' = worker 0 1 where + n = fromIntegral n' + worker j x + | j>n = [] + | True = let j'=j+1 in x : worker j' (div (x*(n-j)) j') + +multinomial :: Integral a => [a] -> Integer +multinomial xs = div + (factorial (sum xs)) + (product [ factorial x | x<-xs ]) + +-------------------------------------------------------------------------------- +-- * Catalan numbers + +-- | Catalan numbers. OEIS:A000108. +catalan :: Integral a => a -> Integer +catalan n + | n < 0 = 0 + | otherwise = binomial (n+n) n `div` fromIntegral (n+1) + +-- | Catalan's triangle. OEIS:A009766. +-- Note: +-- +-- > catalanTriangle n n == catalan n +-- > catalanTriangle n k == countStandardYoungTableaux (toPartition [n,k]) +-- +catalanTriangle :: Integral a => a -> a -> Integer +catalanTriangle n k + | k > n = 0 + | k < 0 = 0 + | otherwise = (binomial (n+k) n * fromIntegral (n-k+1)) `div` fromIntegral (n+1) + +-------------------------------------------------------------------------------- +-- * Stirling numbers + +-- | Rows of (signed) Stirling numbers of the first kind. OEIS:A008275. +-- Coefficients of the polinomial @(x-1)*(x-2)*...*(x-n+1)@. +-- This function uses the recursion formula. +signedStirling1stArray :: Integral a => a -> Array Int Integer +signedStirling1stArray n + | n < 1 = error "stirling1stArray: n should be at least 1" + | n == 1 = listArray (1,1 ) [1] + | otherwise = listArray (1,n') [ lkp (k-1) - fromIntegral (n-1) * lkp k | k<-[1..n'] ] + where + prev = signedStirling1stArray (n-1) + n' = fromIntegral n :: Int + lkp j | j < 1 = 0 + | j >= n' = 0 + | otherwise = prev ! j + +-- | (Signed) Stirling numbers of the first kind. OEIS:A008275. +-- This function uses 'signedStirling1stArray', so it shouldn't be used +-- to compute /many/ Stirling numbers. +-- +-- Argument order: @signedStirling1st n k@ +-- +signedStirling1st :: Integral a => a -> a -> Integer +signedStirling1st n k + | k==0 && n==0 = 1 + | k < 1 = 0 + | k > n = 0 + | otherwise = signedStirling1stArray n ! (fromIntegral k) + +-- | (Unsigned) Stirling numbers of the first kind. See 'signedStirling1st'. +unsignedStirling1st :: Integral a => a -> a -> Integer +unsignedStirling1st n k = abs (signedStirling1st n k) + +-- | Stirling numbers of the second kind. OEIS:A008277. +-- This function uses an explicit formula. +-- +-- Argument order: @stirling2nd n k@ +-- +stirling2nd :: Integral a => a -> a -> Integer +stirling2nd n k + | k==0 && n==0 = 1 + | k < 1 = 0 + | k > n = 0 + | otherwise = sum xs `div` factorial k where + xs = [ negateIfOdd (k-i) $ binomial k i * (fromIntegral i)^n | i<-[0..k] ] + +-------------------------------------------------------------------------------- +-- * Bernoulli numbers + +-- | Bernoulli numbers. @bernoulli 1 == -1%2@ and @bernoulli k == 0@ for +-- k>2 and /odd/. This function uses the formula involving Stirling numbers +-- of the second kind. Numerators: A027641, denominators: A027642. +bernoulli :: Integral a => a -> Rational +bernoulli n + | n < 0 = error "bernoulli: n should be nonnegative" + | n == 0 = 1 + | n == 1 = -1/2 + | otherwise = sum [ f k | k<-[1..n] ] + where + f k = toRational (negateIfOdd (n+k) $ factorial k * stirling2nd n k) + / toRational (k+1) + +-------------------------------------------------------------------------------- +-- * Bell numbers + +-- | Bell numbers (Sloane's A000110) from B(0) up to B(n). B(0)=B(1)=1, B(2)=2, etc. +-- +-- The Bell numbers count the number of /set partitions/ of a set of size @n@ +-- +-- See +-- +bellNumbersArray :: Integral a => a -> Array Int Integer +bellNumbersArray nn = arr where + arr = array (0::Int,n) kvs + n = fromIntegral nn :: Int + kvs = (0,1) : [ (k, f k) | k<-[1..n] ] + f n = sum' [ binomial (n-1) k * arr ! k | k<-[0..n-1] ] + +-- | The n-th Bell number B(n), using the Stirling numbers of the second kind. +-- This may be slower than using 'bellNumbersArray'. +bellNumber :: Integral a => a -> Integer +bellNumber nn + | n < 0 = error "bellNumber: expecting a nonnegative index" + | n == 0 = 1 + | otherwise = sum' [ stirling2nd n k | k<-[1..n] ] + where + n = fromIntegral nn :: Int + +-------------------------------------------------------------------------------- + + + diff --git a/Math/Combinat/Numbers/Primes.hs b/Math/Combinat/Numbers/Primes.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Numbers/Primes.hs @@ -0,0 +1,354 @@ + +-- | Prime numbers and related number theoretical stuff. + +module Math.Combinat.Numbers.Primes + ( -- * List of prime numbers + primes + , primesSimple + , primesTMWE + -- * Prime factorization + , groupIntegerFactors + , integerFactorsTrialDivision + -- * Integer logarithm + , integerLog2 + , ceilingLog2 + -- * Integer square root + , isSquare + , integerSquareRoot + , ceilingSquareRoot + , integerSquareRoot' + , integerSquareRootNewton' + -- * Modulo @m@ arithmetic + , powerMod + -- * Prime testing + , millerRabinPrimalityTest + , isProbablyPrime + , isVeryProbablyPrime + ) + where + +-------------------------------------------------------------------------------- + +-- import Math.Combinat.Numbers + +import Data.List ( group , sort ) +import Data.Bits + +import System.Random + +-------------------------------------------------------------------------------- +-- List of prime numbers + +-- | Infinite list of primes, using the TMWE algorithm. +primes :: [Integer] +primes = primesTMWE + +-- | A relatively simple but still quite fast implementation of list of primes. +-- By Will Ness +primesSimple :: [Integer] +primesSimple = 2 : 3 : sieve 0 primes' 5 where + primes' = tail primesSimple + sieve k (p:ps) x = noDivs k h ++ sieve (k+1) ps (t+2) where + t = p*p + h = [x,x+2..t-2] + noDivs k = filter (\x -> all (\y -> rem x y /= 0) (take k primes')) + +-- | List of primes, using tree merge with wheel. Code by Will Ness. +primesTMWE :: [Integer] +primesTMWE = 2:3:5:7: gaps 11 wheel (fold3t $ roll 11 wheel primes') where + + primes' = 11: gaps 13 (tail wheel) (fold3t $ roll 11 wheel primes') + fold3t ((x:xs): ~(ys:zs:t)) + = x : union xs (union ys zs) `union` fold3t (pairs t) + pairs ((x:xs):ys:t) = (x : union xs ys) : pairs t + wheel = 2:4:2:4:6:2:6:4:2:4:6:6:2:6:4:2:6:4:6:8:4:2:4:2: + 4:8:6:4:6:2:4:6:2:6:6:4:2:4:6:2:6:4:2:4:2:10:2:10:wheel + gaps k ws@(w:t) cs@ ~(c:u) + | k==c = gaps (k+w) t u + | True = k : gaps (k+w) t cs + roll k ws@(w:t) ps@ ~(p:u) + | k==p = scanl (\c d->c+p*d) (p*p) ws : roll (k+w) t u + | True = roll (k+w) t ps + + minus xxs@(x:xs) yys@(y:ys) = case compare x y of + LT -> x : minus xs yys + EQ -> minus xs ys + GT -> minus xxs ys + minus xs [] = xs + minus [] _ = [] + + union xxs@(x:xs) yys@(y:ys) = case compare x y of + LT -> x : union xs yys + EQ -> x : union xs ys + GT -> y : union xxs ys + union xs [] = xs + union [] ys =ys + +-------------------------------------------------------------------------------- +-- Prime factorization + +-- | Groups integer factors. Example: from [2,2,2,3,3,5] we produce [(2,3),(3,2),(5,1)] +groupIntegerFactors :: [Integer] -> [(Integer,Int)] +groupIntegerFactors = map f . group . sort where + f xs = (head xs, length xs) + +-- | The naive trial division algorithm. +integerFactorsTrialDivision :: Integer -> [Integer] +integerFactorsTrialDivision n + | n<1 = error "integerFactorsTrialDivision: n should be at least 1" + | otherwise = go primes n + where + go _ 1 = [] + go rs k = sub ps k where + sub [] k = [k] + sub qqs@(q:qs) k = case mod k q of + 0 -> q : go qqs (div k q) + _ -> sub qs k + ps = takeWhile (\p -> p*p <= k) rs +{- + go 1 = [] + go k = sub ps k where + sub [] k = [k] + sub (q:qs) k = case mod k q of + 0 -> q : go (div k q) + _ -> sub qs k + ps = takeWhile (\p -> p*p <= k) primes +-} + +{- +-- brute force testing of factors +ifactorsTest :: (Integer -> [Integer]) -> Integer -> Bool +ifactorsTest alg n = and [ product (alg k) == k | k<-[1..n] ] +-} + +-------------------------------------------------------------------------------- +-- Integer logarithm + +-- | Largest integer @k@ such that @2^k@ is smaller or equal to @n@ +integerLog2 :: Integer -> Integer +integerLog2 n = go n where + go 0 = -1 + go k = 1 + go (shiftR k 1) + +-- | Smallest integer @k@ such that @2^k@ is larger or equal to @n@ +ceilingLog2 :: Integer -> Integer +ceilingLog2 0 = 0 +ceilingLog2 n = 1 + go (n-1) where + go 0 = -1 + go k = 1 + go (shiftR k 1) + +-------------------------------------------------------------------------------- +-- Integer square root + +isSquare :: Integer -> Bool +isSquare n = + if (fromIntegral $ mod n 32) `elem` rs + then snd (integerSquareRoot' n) == 0 + else False + where + rs = [0,1,4,9,16,17,25] :: [Int] + +-- | Integer square root (largest integer whose square is smaller or equal to the input) +-- using Newton's method, with a faster (for large numbers) inital guess based on bit shifts. +integerSquareRoot :: Integer -> Integer +integerSquareRoot = fst . integerSquareRoot' + +-- | Smallest integer whose square is larger or equal to the input +ceilingSquareRoot :: Integer -> Integer +ceilingSquareRoot n = (if r>0 then u+1 else u) where (u,r) = integerSquareRoot' n + +-- | We also return the excess residue; that is +-- +-- > (a,r) = integerSquareRoot' n +-- +-- means that +-- +-- > a*a + r = n +-- > a*a <= n < (a+1)*(a+1) +integerSquareRoot' :: Integer -> (Integer,Integer) +integerSquareRoot' n + | n<0 = error "integerSquareRoot: negative input" + | n<2 = (n,0) + | otherwise = go firstGuess + where + k = integerLog2 n + firstGuess = 2^(div (k+2) 2) -- !! note that (div (k+1) 2) is NOT enough !! + go a = + if m < a + then go a' + else (a, r + a*(m-a)) + where + (m,r) = divMod n a + a' = div (m + a) 2 + +-- | Newton's method without an initial guess. For very small numbers (<10^10) it +-- is somewhat faster than the above version. +integerSquareRootNewton' :: Integer -> (Integer,Integer) +integerSquareRootNewton' n + | n<0 = error "integerSquareRootNewton: negative input" + | n<2 = (n,0) + | otherwise = go (div n 2) + where + go a = + if m < a + then go a' + else (a, r + a*(m-a)) + where + (m,r) = divMod n a + a' = div (m + a) 2 + +{- +-- brute force test of integer square root +isqrt_test n1 n2 = + [ k + | k<-[n1..n2] + , let (a,r) = integerSquareRoot' k + , (a*a+r/=k) || (a*a>k) || (a+1)*(a+1)<=k + ] +-} + +-------------------------------------------------------------------------------- +-- Modulo @m@ arithmetic + +-- | Efficient powers modulo m. +-- +-- > powerMod a k m == (a^k) `mod` m +powerMod :: Integer -> Integer -> Integer -> Integer +powerMod a' k m = {- debug bs $ -} go a bs where + + bs = bin k + + bin 0 = [] + bin x = (x .&. 1 /= 0) : bin (shiftR x 1) + + a = mod a' m + + go _ [] = 1 + go x (b:bs) = -- debug (x,b) $ + if b + then mod (x*rest) m + else rest + where + rest = go (mod (x*x) m) bs + +-------------------------------------------------------------------------------- +-- Prime testing + +-- | Miller-Rabin Primality Test (taken from Haskell wiki). +-- We test the primality of the first argument @n@ by using the second argument @a@ as a candidate witness. +-- If it returs @False@, then @n@ is composite. If it returns @True@, then @n@ is either prime or composite. +-- +-- A random choice between @2@ and @(n-2)@ is a good choice for @a@. +millerRabinPrimalityTest :: Integer -> Integer -> Bool +millerRabinPrimalityTest n a + | a <= 1 || a >= n-1 = + error $ "millerRabinPrimalityTest: a out of range (" ++ show a ++ " for "++ show n ++ ")" + | n < 2 = False + | even n = False + | b0 == 1 || b0 == n' = True + | otherwise = iter (tail b) + where + n' = n-1 + (k,m) = find2km n' + b0 = powMod n a m + b = take (fromIntegral k) $ iterate (squareMod n) b0 + iter [] = False + iter (x:xs) + | x == 1 = False + | x == n' = True + | otherwise = iter xs + + +{-# SPECIALIZE find2km :: Integer -> (Integer,Integer) #-} +find2km :: Integral a => a -> (a,a) +find2km n = f 0 n where + f k m + | r == 1 = (k,m) + | otherwise = f (k+1) q + where (q,r) = quotRem m 2 + +{-# SPECIALIZE pow' :: (Integer -> Integer -> Integer) -> (Integer -> Integer) -> Integer -> Integer -> Integer #-} +pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a +pow' _ _ _ 0 = 1 +pow' mul sq x' n' = f x' n' 1 where + f x n y + | n == 1 = x `mul` y + | r == 0 = f x2 q y + | otherwise = f x2 q (x `mul` y) + where + (q,r) = quotRem n 2 + x2 = sq x + +{-# SPECIALIZE mulMod :: Integer -> Integer -> Integer -> Integer #-} +mulMod :: Integral a => a -> a -> a -> a +mulMod a b c = (b * c) `mod` a + +{-# SPECIALIZE squareMod :: Integer -> Integer -> Integer #-} +squareMod :: Integral a => a -> a -> a +squareMod a b = (b * b) `rem` a + +{-# SPECIALIZE powMod :: Integer -> Integer -> Integer -> Integer #-} +powMod :: Integral a => a -> a -> a -> a +powMod m = pow' (mulMod m) (squareMod m) + +-------------------------------------------------------------------------------- + +-- | For very small numbers, we use trial division, for larger numbers, we apply the +-- Miller-Rabin primality test @log4(n)@ times, with candidate witnesses derived +-- deterministically from @n@ using a pseudo-random sequence +-- (which /should be/ based on a cryptographic hash function, but isn\'t, yet). +-- +-- Thus the candidate witnesses should behave essentially like random, but the +-- resulting function is still a deterministic, pure function. +-- +-- TODO: implement the hash sequence, at the moment we use 'System.Random' instead... +-- +isProbablyPrime :: Integer -> Bool +isProbablyPrime n + | n < 2 = False + | even n = (n==2) + | n < 1000 = length (integerFactorsTrialDivision n) == 1 + | otherwise = and [ millerRabinPrimalityTest n a | a <- witnessList ] + where + log2n = integerLog2 n + nchecks = 1 + fromInteger (div log2n 2) :: Int + witnessList = take nchecks pseudoRnds + pseudoRnds = 2 : [ a | a <- integerRndSequence n , a > 1 && a < (n-1) ] + +-- | A more exhaustive version of 'isProbablyPrime', this one tests candidate +-- witnesses both the first log4(n) prime numbers and then log4(n) pseudo-random +-- numbers +isVeryProbablyPrime :: Integer -> Bool +isVeryProbablyPrime n + | n < 2 = False + | even n = (n==2) + | n < 1000 = length (integerFactorsTrialDivision n) == 1 + | otherwise = and [ millerRabinPrimalityTest n a | a <- witnessList ] + where + log2n = integerLog2 n + nchecks = 1 + fromInteger (div log2n 2) :: Int + witnessList = take nchecks primes ++ take nchecks pseudoRnds + pseudoRnds = [ a | a <- integerRndSequence (n+3) , a > 1 && a < (n-1) ] + +-------------------------------------------------------------------------------- + +{- +-- | Given an integer @n@, we return an infinite sequence of pseudo-random integers +-- between @0..n-1@, generated using a crypographic hash function. +-- +integerHashSequence :: Integer -> [Integer] +integerHashSequence = error "integerHashSequence: not implemented yet" +-} + +-- | Given an integer @n@, we initialize a system random generator with using a +-- seed derived from @n@ (note that this uses at most 32 or 64 bits), and generate +-- an infinite sequence of pseudo-random integers between @0..n-1@, generated by +-- that random generator. +-- +-- Note that this is not really a preferred way of generating such sequences! +-- +integerRndSequence :: Integer -> [Integer] +integerRndSequence n = randomRs (0,n-1) gen where + gen = mkStdGen $ fromInteger (n + 17 * integerLog2 n) + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Numbers/Series.hs b/Math/Combinat/Numbers/Series.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Numbers/Series.hs @@ -0,0 +1,376 @@ + +-- | Some basic univariate power series expansions. +-- This module is not re-exported by "Math.Combinat". +-- +-- Note: the \"@convolveWithXXX@\" functions are much faster than the equivalent +-- @(XXX \`convolve\`)@! +-- +-- TODO: better names for these functions. +-- + +{-# LANGUAGE CPP, GeneralizedNewtypeDeriving #-} +module Math.Combinat.Numbers.Series where + +-------------------------------------------------------------------------------- + +import Data.List + +import Math.Combinat.Sign +import Math.Combinat.Numbers +import Math.Combinat.Partitions.Integer +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * Trivial series + +-- | The series [1,0,0,0,0,...], which is the neutral element for the convolution. +{-# SPECIALIZE unitSeries :: [Integer] #-} +unitSeries :: Num a => [a] +unitSeries = 1 : repeat 0 + +-- | Constant zero series +zeroSeries :: Num a => [a] +zeroSeries = repeat 0 + +-- | Power series representing a constant function +constSeries :: Num a => a -> [a] +constSeries x = x : repeat 0 + +-- | The power series representation of the identity function @x@ +idSeries :: Num a => [a] +idSeries = 0 : 1 : repeat 0 + +-- | The power series representation of @x^n@ +powerTerm :: Num a => Int -> [a] +powerTerm n = replicate n 0 ++ (1 : repeat 0) + +-------------------------------------------------------------------------------- +-- * Basic operations on power series + +addSeries :: Num a => [a] -> [a] -> [a] +addSeries xs ys = longZipWith 0 0 (+) xs ys + +sumSeries :: Num a => [[a]] -> [a] +sumSeries [] = [0] +sumSeries xs = foldl1' addSeries xs + +subSeries :: Num a => [a] -> [a] -> [a] +subSeries xs ys = longZipWith 0 0 (-) xs ys + +negateSeries :: Num a => [a] -> [a] +negateSeries = map negate + +scaleSeries :: Num a => a -> [a] -> [a] +scaleSeries s = map (*s) + +mulSeries :: Num a => [a] -> [a] -> [a] +mulSeries = convolve + +productOfSeries :: Num a => [[a]] -> [a] +productOfSeries = convolveMany + +-------------------------------------------------------------------------------- +-- * Convolution (product) + +-- | Convolution of series (that is, multiplication of power series). +-- The result is always an infinite list. Warning: This is slow! +convolve :: Num a => [a] -> [a] -> [a] +convolve xs1 ys1 = res where + res = [ foldl' (+) 0 (zipWith (*) xs (reverse (take n ys))) + | n<-[1..] + ] + xs = xs1 ++ repeat 0 + ys = ys1 ++ repeat 0 + +-- | Convolution (= product) of many series. Still slow! +convolveMany :: Num a => [[a]] -> [a] +convolveMany [] = 1 : repeat 0 +convolveMany xss = foldl1 convolve xss + +-------------------------------------------------------------------------------- +-- * Reciprocals of general power series + +-- | Given a power series, we iteratively compute its multiplicative inverse +reciprocalSeries :: (Eq a, Fractional a) => [a] -> [a] +reciprocalSeries series = case series of + [] -> error "reciprocalSeries: empty input series (const 0 function does not have an inverse)" + (a:as) -> case a of + 0 -> error "reciprocalSeries: input series has constant term 0" + _ -> map (/a) $ integralReciprocalSeries $ map (/a) series + +-- | Given a power series starting with @1@, we can compute its multiplicative inverse +-- without divisions. +-- +{-# SPECIALIZE integralReciprocalSeries :: [Int] -> [Int] #-} +{-# SPECIALIZE integralReciprocalSeries :: [Integer] -> [Integer] #-} +integralReciprocalSeries :: (Eq a, Num a) => [a] -> [a] +integralReciprocalSeries series = case series of + [] -> error "integralReciprocalSeries: empty input series (const 0 function does not have an inverse)" + (a:as) -> case a of + 1 -> 1 : worker [1] + _ -> error "integralReciprocalSeries: input series must start with 1" + where + worker bs = let b' = - sum (zipWith (*) (tail series) bs) + in b' : worker (b':bs) + +-------------------------------------------------------------------------------- +-- * Composition of formal power series + +-- | @g \`composeSeries\` f@ is the power series expansion of @g(f(x))@. +-- This is a synonym for @flip substitute@. +-- +-- We require that the constant term of @f@ is zero. +composeSeries :: (Eq a, Num a) => [a] -> [a] -> [a] +composeSeries g f = substitute f g + +-- | @substitute f g@ is the power series corresponding to @g(f(x))@. +-- Equivalently, this is the composition of univariate functions (in the \"wrong\" order). +-- +-- Note: for this to be meaningful in general (not depending on convergence properties), +-- we need that the constant term of @f@ is zero. +substitute :: (Eq a, Num a) => [a] -> [a] -> [a] +substitute as_ bs_ = + case head as of + 0 -> [ f n | n<-[0..] ] + _ -> error "PowerSeries/substitute: we expect the the constant term of the inner series to be zero" + where + as = as_ ++ repeat 0 + bs = bs_ ++ repeat 0 + a i = as !! i + b j = bs !! j + f n = sum + [ b m * product [ (a i)^j | (i,j)<-es ] * fromInteger (multinomial (map snd es)) + | p <- partitions n + , let es = toExponentialForm p + , let m = partitionWidth p + ] + +-------------------------------------------------------------------------------- +-- * Lagrange inversions + +-- | Coefficients of the Lagrange inversion +lagrangeCoeff :: Partition -> Integer +lagrangeCoeff p = div numer denom where + numer = (-1)^m * product (map fromIntegral [n+1..n+m]) + denom = fromIntegral (n+1) * product (map (factorial . snd) es) + m = partitionWidth p + n = partitionWeight p + es = toExponentialForm p + +-- | We expect the input series to match @(0:1:_)@. The following is true for the result (at least with exact arithmetic): +-- +-- > substitute f (integralLagrangeInversion f) == (0 : 1 : repeat 0) +-- > substitute (integralLagrangeInversion f) f == (0 : 1 : repeat 0) +-- +integralLagrangeInversion :: (Eq a, Num a) => [a] -> [a] +integralLagrangeInversion series_ = + case series of + (0:1:rest) -> 0 : 1 : [ f n | n<-[1..] ] + _ -> error "integralLagrangeInversion: the series should start with (0 + x + a2*x^2 + ...)" + where + series = series_ ++ repeat 0 + as = tail series + a i = as !! i + f n = sum [ fromInteger (lagrangeCoeff p) * product [ (a i)^j | (i,j) <- toExponentialForm p ] + | p <- partitions n + ] + +-- | We expect the input series to match @(0:a1:_)@. with a1 nonzero The following is true for the result (at least with exact arithmetic): +-- +-- > substitute f (lagrangeInversion f) == (0 : 1 : repeat 0) +-- > substitute (lagrangeInversion f) f == (0 : 1 : repeat 0) +-- +lagrangeInversion :: (Eq a, Fractional a) => [a] -> [a] +lagrangeInversion series_ = + case series of + (0:a1:rest) -> if a1 ==0 + then err + else 0 : (1/a1) : [ f n / a1^(n+1) | n<-[1..] ] + _ -> err + where + err = error "lagrangeInversion: the series should start with (0 + a1*x + a2*x^2 + ...) where a1 is non-zero" + series = series_ ++ repeat 0 + a1 = series !! 1 + as = map (/a1) (tail series) + a i = as !! i + f n = sum [ fromInteger (lagrangeCoeff p) * product [ (a i)^j | (i,j) <- toExponentialForm p ] + | p <- partitions n + ] + +-------------------------------------------------------------------------------- +-- * Power series expansions of elementary functions + +-- | Power series expansion of @exp(x)@ +expSeries :: Fractional a => [a] +expSeries = go 0 1 where + go i e = e : go (i+1) (e / (i+1)) + +-- | Power series expansion of @cos(x)@ +cosSeries :: Fractional a => [a] +cosSeries = go 0 1 where + go i e = e : 0 : go (i+2) (-e / ((i+1)*(i+2))) + +-- | Power series expansion of @sin(x)@ +sinSeries :: Fractional a => [a] +sinSeries = go 1 1 where + go i e = 0 : e : go (i+2) (-e / ((i+1)*(i+2))) + +-- | Power series expansion of @cosh(x)@ +coshSeries :: Fractional a => [a] +coshSeries = go 0 1 where + go i e = e : 0 : go (i+2) (e / ((i+1)*(i+2))) + +-- | Power series expansion of @sinh(x)@ +sinhSeries :: Fractional a => [a] +sinhSeries = go 1 1 where + go i e = 0 : e : go (i+2) (e / ((i+1)*(i+2))) + +-- | Power series expansion of @log(1+x)@ +log1Series :: Fractional a => [a] +log1Series = 0 : go 1 1 where + go i e = (e/i) : go (i+1) (-e) + +-- | Power series expansion of @(1-Sqrt[1-4x])/(2x)@ (the coefficients are the Catalan numbers) +dyckSeries :: Num a => [a] +dyckSeries = [ fromInteger (catalan i) | i<-[(0::Int)..] ] + +-------------------------------------------------------------------------------- +-- * \"Coin\" series + +-- | Power series expansion of +-- +-- > 1 / ( (1-x^k_1) * (1-x^k_2) * ... * (1-x^k_n) ) +-- +-- Example: +-- +-- @(coinSeries [2,3,5])!!k@ is the number of ways +-- to pay @k@ dollars with coins of two, three and five dollars. +-- +-- TODO: better name? +coinSeries :: [Int] -> [Integer] +coinSeries [] = 1 : repeat 0 +coinSeries (k:ks) = xs where + xs = zipWith (+) (coinSeries ks) (replicate k 0 ++ xs) + +-- | Generalization of the above to include coefficients: expansion of +-- +-- > 1 / ( (1-a_1*x^k_1) * (1-a_2*x^k_2) * ... * (1-a_n*x^k_n) ) +-- +coinSeries' :: Num a => [(a,Int)] -> [a] +coinSeries' [] = 1 : repeat 0 +coinSeries' ((a,k):aks) = xs where + xs = zipWith (+) (coinSeries' aks) (replicate k 0 ++ map (*a) xs) + +convolveWithCoinSeries :: [Int] -> [Integer] -> [Integer] +convolveWithCoinSeries ks series1 = worker ks where + series = series1 ++ repeat 0 + worker [] = series + worker (k:ks) = xs where + xs = zipWith (+) (worker ks) (replicate k 0 ++ xs) + +convolveWithCoinSeries' :: Num a => [(a,Int)] -> [a] -> [a] +convolveWithCoinSeries' ks series1 = worker ks where + series = series1 ++ repeat 0 + worker [] = series + worker ((a,k):aks) = xs where + xs = zipWith (+) (worker aks) (replicate k 0 ++ map (*a) xs) + +-------------------------------------------------------------------------------- +-- * Reciprocals of products of polynomials + +-- | Convolution of many 'pseries', that is, the expansion of the reciprocal +-- of a product of polynomials +productPSeries :: [[Int]] -> [Integer] +productPSeries = foldl (flip convolveWithPSeries) unitSeries + +-- | The same, with coefficients. +productPSeries' :: Num a => [[(a,Int)]] -> [a] +productPSeries' = foldl (flip convolveWithPSeries') unitSeries + +convolveWithProductPSeries :: [[Int]] -> [Integer] -> [Integer] +convolveWithProductPSeries kss ser = foldl (flip convolveWithPSeries) ser kss + +-- | This is the most general function in this module; all the others +-- are special cases of this one. +convolveWithProductPSeries' :: Num a => [[(a,Int)]] -> [a] -> [a] +convolveWithProductPSeries' akss ser = foldl (flip convolveWithPSeries') ser akss + +-------------------------------------------------------------------------------- +-- * Reciprocals of polynomials + +-- Reciprocals of polynomials, without coefficients + +-- | The power series expansion of +-- +-- > 1 / (1 - x^k_1 - x^k_2 - x^k_3 - ... - x^k_n) +-- +pseries :: [Int] -> [Integer] +pseries ks = convolveWithPSeries ks unitSeries + +-- | Convolve with (the expansion of) +-- +-- > 1 / (1 - x^k_1 - x^k_2 - x^k_3 - ... - x^k_n) +-- +convolveWithPSeries :: [Int] -> [Integer] -> [Integer] +convolveWithPSeries ks series1 = ys where + series = series1 ++ repeat 0 + ys = worker ks ys + worker [] _ = series + worker (k:ks) ys = xs where + xs = zipWith (+) (replicate k 0 ++ ys) (worker ks ys) + +-------------------------------------------------------------------------------- +-- Reciprocals of polynomials, with coefficients + +-- | The expansion of +-- +-- > 1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3 - ... - a_n*x^k_n) +-- +pseries' :: Num a => [(a,Int)] -> [a] +pseries' aks = convolveWithPSeries' aks unitSeries + +-- | Convolve with (the expansion of) +-- +-- > 1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3 - ... - a_n*x^k_n) +-- +convolveWithPSeries' :: Num a => [(a,Int)] -> [a] -> [a] +convolveWithPSeries' aks series1 = ys where + series = series1 ++ repeat 0 + ys = worker aks ys + worker [] _ = series + worker ((a,k):aks) ys = xs where + xs = zipWith (+) (replicate k 0 ++ map (*a) ys) (worker aks ys) + +{- +data Sign = Plus | Minus deriving (Eq,Show) + +signValue :: Num a => Sign -> a +signValue Plus = 1 +signValue Minus = -1 +-} + +signedPSeries :: [(Sign,Int)] -> [Integer] +signedPSeries aks = convolveWithSignedPSeries aks unitSeries + +-- | Convolve with (the expansion of) +-- +-- > 1 / (1 +- x^k_1 +- x^k_2 +- x^k_3 +- ... +- x^k_n) +-- +-- Should be faster than using `convolveWithPSeries'`. +-- Note: 'Plus' corresponds to the coefficient @-1@ in `pseries'` (since +-- there is a minus sign in the definition there)! +convolveWithSignedPSeries :: [(Sign,Int)] -> [Integer] -> [Integer] +convolveWithSignedPSeries aks series1 = ys where + series = series1 ++ repeat 0 + ys = worker aks ys + worker [] _ = series + worker ((a,k):aks) ys = xs where + xs = case a of + Minus -> zipWith (+) one two + Plus -> zipWith (-) one two + one = worker aks ys + two = replicate k 0 ++ ys + +-------------------------------------------------------------------------------- + + diff --git a/Math/Combinat/Partitions.hs b/Math/Combinat/Partitions.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions.hs @@ -0,0 +1,22 @@ + +-- | Partitions of integers and multisets. +-- Integer partitions are nonincreasing sequences of positive integers. +-- +-- See: +-- +-- * Donald E. Knuth: The Art of Computer Programming, vol 4, pre-fascicle 3B. +-- +-- * +-- + +{-# LANGUAGE BangPatterns #-} +module Math.Combinat.Partitions + ( module Math.Combinat.Partitions.Integer + ) + where + +-------------------------------------------------------------------------------- + +import Math.Combinat.Partitions.Integer + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Partitions/Integer.hs b/Math/Combinat/Partitions/Integer.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/Integer.hs @@ -0,0 +1,708 @@ + +-- | Partitions of integers. +-- Integer partitions are nonincreasing sequences of positive integers. +-- +-- See: +-- +-- * Donald E. Knuth: The Art of Computer Programming, vol 4, pre-fascicle 3B. +-- +-- * +-- +-- For example the partition +-- +-- > Partition [8,6,3,3,1] +-- +-- can be represented by the (English notation) Ferrers diagram: +-- +-- <> +-- + +{-# LANGUAGE CPP, BangPatterns, ScopedTypeVariables #-} +module Math.Combinat.Partitions.Integer where + +-------------------------------------------------------------------------------- + +import Data.List +import Control.Monad ( liftM , replicateM ) + +-- import Data.Map (Map) +-- import qualified Data.Map as Map + +import Math.Combinat.Classes +import Math.Combinat.ASCII as ASCII +import Math.Combinat.Numbers (factorial,binomial,multinomial) +import Math.Combinat.Helper + +import Data.Array +import System.Random + +-------------------------------------------------------------------------------- +-- * Type and basic stuff + +-- | A partition of an integer. The additional invariant enforced here is that partitions +-- are monotone decreasing sequences of /positive/ integers. The @Ord@ instance is lexicographical. +newtype Partition = Partition [Int] deriving (Eq,Ord,Show,Read) + +instance HasNumberOfParts Partition where + numberOfParts (Partition p) = length p + +--------------------------------------------------------------------------------- + +-- | Sorts the input, and cuts the nonpositive elements. +mkPartition :: [Int] -> Partition +mkPartition xs = Partition $ sortBy (reverseCompare) $ filter (>0) xs + +-- | Assumes that the input is decreasing. +toPartitionUnsafe :: [Int] -> Partition +toPartitionUnsafe = Partition + +-- | Checks whether the input is an integer partition. See the note at 'isPartition'! +toPartition :: [Int] -> Partition +toPartition xs = if isPartition xs + then toPartitionUnsafe xs + else error "toPartition: not a partition" + +-- | This returns @True@ if the input is non-increasing sequence of +-- /positive/ integers (possibly empty); @False@ otherwise. +-- +isPartition :: [Int] -> Bool +isPartition [] = True +isPartition [x] = x > 0 +isPartition (x:xs@(y:_)) = (x >= y) && isPartition xs + +isEmptyPartition :: Partition -> Bool +isEmptyPartition (Partition p) = null p + +emptyPartition :: Partition +emptyPartition = Partition [] + +instance CanBeEmpty Partition where + empty = emptyPartition + isEmpty = isEmptyPartition + +fromPartition :: Partition -> [Int] +fromPartition (Partition part) = part + +-- | The first element of the sequence. +partitionHeight :: Partition -> Int +partitionHeight (Partition part) = case part of + (p:_) -> p + [] -> 0 + +-- | The length of the sequence (that is, the number of parts). +partitionWidth :: Partition -> Int +partitionWidth (Partition part) = length part + +instance HasHeight Partition where + height = partitionHeight + +instance HasWidth Partition where + width = partitionWidth + +heightWidth :: Partition -> (Int,Int) +heightWidth part = (height part, width part) + +-- | The weight of the partition +-- (that is, the sum of the corresponding sequence). +partitionWeight :: Partition -> Int +partitionWeight (Partition part) = sum' part + +instance HasWeight Partition where + weight = partitionWeight + +-- | The dual (or conjugate) partition. +dualPartition :: Partition -> Partition +dualPartition (Partition part) = Partition (_dualPartition part) + +instance HasDuality Partition where + dual = dualPartition + +data Pair = Pair !Int !Int + +_dualPartition :: [Int] -> [Int] +_dualPartition [] = [] +_dualPartition xs = go 0 (diffSequence xs) [] where + go !i (d:ds) acc = go (i+1) ds (d:acc) + go n [] acc = finish n acc + finish !j (k:ks) = replicate k j ++ finish (j-1) ks + finish _ [] = [] + +{- +-- more variations: + +_dualPartition_b :: [Int] -> [Int] +_dualPartition_b [] = [] +_dualPartition_b xs = go 1 (diffSequence xs) [] where + go !i (d:ds) acc = go (i+1) ds ((d,i):acc) + go _ [] acc = concatMap (\(d,i) -> replicate d i) acc + +_dualPartition_c :: [Int] -> [Int] +_dualPartition_c [] = [] +_dualPartition_c xs = reverse $ concat $ zipWith f [1..] (diffSequence xs) where + f _ 0 = [] + f k d = replicate d k +-} + +-- | A simpler, but bit slower (about twice?) implementation of dual partition +_dualPartitionNaive :: [Int] -> [Int] +_dualPartitionNaive [] = [] +_dualPartitionNaive xs@(k:_) = [ length $ filter (>=i) xs | i <- [1..k] ] + +-- | From a sequence @[a1,a2,..,an]@ computes the sequence of differences +-- @[a1-a2,a2-a3,...,an-0]@ +diffSequence :: [Int] -> [Int] +diffSequence = go where + go (x:ys@(y:_)) = (x-y) : go ys + go [x] = [x] + go [] = [] + +-- | Example: +-- +-- > elements (toPartition [5,4,1]) == +-- > [ (1,1), (1,2), (1,3), (1,4), (1,5) +-- > , (2,1), (2,2), (2,3), (2,4) +-- > , (3,1) +-- > ] +-- +elements :: Partition -> [(Int,Int)] +elements (Partition part) = _elements part + +_elements :: [Int] -> [(Int,Int)] +_elements shape = [ (i,j) | (i,l) <- zip [1..] shape, j<-[1..l] ] + +--------------------------------------------------------------------------------- +-- * Exponential form + +-- | We convert a partition to exponential form. +-- @(i,e)@ mean @(i^e)@; for example @[(1,4),(2,3)]@ corresponds to @(1^4)(2^3) = [2,2,2,1,1,1,1]@. Another example: +-- +-- > toExponentialForm (Partition [5,5,3,2,2,2,2,1,1]) == [(1,2),(2,4),(3,1),(5,2)] +-- +toExponentialForm :: Partition -> [(Int,Int)] +toExponentialForm = _toExponentialForm . fromPartition + +_toExponentialForm :: [Int] -> [(Int,Int)] +_toExponentialForm = reverse . map (\xs -> (head xs,length xs)) . group + +fromExponentialFrom :: [(Int,Int)] -> Partition +fromExponentialFrom = Partition . sortBy reverseCompare . go where + go ((j,e):rest) = replicate e j ++ go rest + go [] = [] + +--------------------------------------------------------------------------------- +-- * Automorphisms + +-- | Computes the number of \"automorphisms\" of a given integer partition. +countAutomorphisms :: Partition -> Integer +countAutomorphisms = _countAutomorphisms . fromPartition + +_countAutomorphisms :: [Int] -> Integer +_countAutomorphisms = multinomial . map length . group + +--------------------------------------------------------------------------------- +-- * Generating partitions + +-- | Partitions of @d@. +partitions :: Int -> [Partition] +partitions = map Partition . _partitions + +-- | Partitions of @d@, as lists +_partitions :: Int -> [[Int]] +_partitions d = go d d where + go _ 0 = [[]] + go !h !n = [ a:as | a<-[1..min n h], as <- go a (n-a) ] + +-- | Number of partitions of @n@ +countPartitions :: Int -> Integer +countPartitions n = partitionCountList !! n + +-- | This uses 'countPartitions'', and thus is slow +countPartitionsNaive :: Int -> Integer +countPartitionsNaive d = countPartitions' (d,d) d + +-------------------------------------------------------------------------------- + +-- | Infinite list of number of partitions of @0,1,2,...@ +-- +-- This uses the infinite product formula the generating function of partitions, recursively +-- expanding it; it is quite fast. +-- +-- > partitionCountList == map countPartitions [0..] +-- +partitionCountList :: [Integer] +partitionCountList = final where + + final = go 1 (1:repeat 0) + + go !k (x:xs) = x : go (k+1) ys where + ys = zipWith (+) xs (take k final ++ ys) + -- explanation: + -- xs == drop k $ f (k-1) + -- ys == drop k $ f (k ) + +{- + +Full explanation of 'partitionCountList': +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +let f k = productPSeries $ map (:[]) [1..k] + +f 0 = [1,0,0,0,0,0,0,0...] +f 1 = [1,1,1,1,1,1,1,1...] +f 2 = [1,1,2,2,3,3,4,4...] +f 3 = [1,1,2,3,4,5,7,8...] + +observe: + +* take (k+1) (f k) == take (k+1) partitionCountList +* f (k+1) == zipWith (+) (f k) (replicate (k+1) 0 ++ f (k+1)) + +now apply (drop (k+1)) to the second one : + +* drop (k+1) (f (k+1)) == zipWith (+) (drop (k+1) $ f k) (f (k+1)) +* f (k+1) = take (k+1) final ++ drop (k+1) (f (k+1)) + +-} + +-------------------------------------------------------------------------------- + +-- | Naive infinite list of number of partitions of @0,1,2,...@ +-- +-- > partitionCountListNaive == map countPartitionsNaive [0..] +-- +-- This is much slower than the power series expansion above. +-- +partitionCountListNaive :: [Integer] +partitionCountListNaive = map countPartitionsNaive [0..] + +-- | All integer partitions up to a given degree (that is, all integer partitions whose sum is less or equal to @d@) +allPartitions :: Int -> [Partition] +allPartitions d = concat [ partitions i | i <- [0..d] ] + +-- | All integer partitions up to a given degree (that is, all integer partitions whose sum is less or equal to @d@), +-- grouped by weight +allPartitionsGrouped :: Int -> [[Partition]] +allPartitionsGrouped d = [ partitions i | i <- [0..d] ] + +-- | All integer partitions fitting into a given rectangle. +allPartitions' + :: (Int,Int) -- ^ (height,width) + -> [Partition] +allPartitions' (h,w) = concat [ partitions' (h,w) i | i <- [0..d] ] where d = h*w + +-- | All integer partitions fitting into a given rectangle, grouped by weight. +allPartitionsGrouped' + :: (Int,Int) -- ^ (height,width) + -> [[Partition]] +allPartitionsGrouped' (h,w) = [ partitions' (h,w) i | i <- [0..d] ] where d = h*w + +-- | # = \\binom { h+w } { h } +countAllPartitions' :: (Int,Int) -> Integer +countAllPartitions' (h,w) = + binomial (h+w) (min h w) + --sum [ countPartitions' (h,w) i | i <- [0..d] ] where d = h*w + +countAllPartitions :: Int -> Integer +countAllPartitions d = sum' [ countPartitions i | i <- [0..d] ] + +-- | Integer partitions of @d@, fitting into a given rectangle, as lists. +_partitions' + :: (Int,Int) -- ^ (height,width) + -> Int -- ^ d + -> [[Int]] +_partitions' _ 0 = [[]] +_partitions' ( 0 , _) d = if d==0 then [[]] else [] +_partitions' ( _ , 0) d = if d==0 then [[]] else [] +_partitions' (!h ,!w) d = + [ i:xs | i <- [1..min d h] , xs <- _partitions' (i,w-1) (d-i) ] + +-- | Partitions of d, fitting into a given rectangle. The order is again lexicographic. +partitions' + :: (Int,Int) -- ^ (height,width) + -> Int -- ^ d + -> [Partition] +partitions' hw d = map toPartitionUnsafe $ _partitions' hw d + +countPartitions' :: (Int,Int) -> Int -> Integer +countPartitions' _ 0 = 1 +countPartitions' (0,_) d = if d==0 then 1 else 0 +countPartitions' (_,0) d = if d==0 then 1 else 0 +countPartitions' (h,w) d = sum + [ countPartitions' (i,w-1) (d-i) | i <- [1..min d h] ] + + +--------------------------------------------------------------------------------- +-- * Random partitions + +-- | Uniformly random partition of the given weight. +-- +-- NOTE: This algorithm is effective for small @n@-s (say @n@ up to a few hundred \/ one thousand it should work nicely), +-- and the first time it is executed may be slower (as it needs to build the table 'partitionCountList' first) +-- +-- Algorithm of Nijenhuis and Wilf (1975); see +-- +-- * Knuth Vol 4A, pre-fascicle 3B, exercise 47; +-- +-- * Nijenhuis and Wilf: Combinatorial Algorithms for Computers and Calculators, chapter 10 +-- +randomPartition :: RandomGen g => Int -> g -> (Partition, g) +randomPartition n g = (p, g') where + ([p], g') = randomPartitions 1 n g + +-- | Generates several uniformly random partitions of @n@ at the same time. +-- Should be a little bit faster then generating them individually. +-- +randomPartitions + :: forall g. RandomGen g + => Int -- ^ number of partitions to generate + -> Int -- ^ the weight of the partitions + -> g -> ([Partition], g) +randomPartitions howmany n = runRand $ replicateM howmany (worker n []) where + + table = listArray (0,n) $ take (n+1) partitionCountList :: Array Int Integer + cnt k = table ! k + + finish :: [(Int,Int)] -> Partition + finish = mkPartition . concatMap f where f (j,d) = replicate j d + + fi :: Int -> Integer + fi = fromIntegral + + find_jd :: Int -> Integer -> (Int,Int) + find_jd m capm = go 0 [ (j,d) | j<-[1..n], d<-[1..div m j] ] where + go :: Integer -> [(Int,Int)] -> (Int,Int) + go !s [] = (1,1) -- ?? + go !s [jd] = jd -- ?? + go !s (jd@(j,d):rest) = + if s' > capm + then jd + else go s' rest + where + s' = s + fi d * cnt (m - j*d) + + worker :: Int -> [(Int,Int)] -> Rand g Partition + worker 0 acc = return $ finish acc + worker !m acc = do + capm <- randChoose (0, (fi m) * cnt m - 1) + let jd@(!j,!d) = find_jd m capm + worker (m - j*d) (jd:acc) + + +--------------------------------------------------------------------------------- +-- * Dominance order + +-- | @q \`dominates\` p@ returns @True@ if @q >= p@ in the dominance order of partitions +-- (this is partial ordering on the set of partitions of @n@). +-- +-- See +-- +dominates :: Partition -> Partition -> Bool +dominates (Partition qs) (Partition ps) + = and $ zipWith (>=) (sums (qs ++ repeat 0)) (sums ps) + where + sums = scanl (+) 0 + + +-- | Lists all partitions of the same weight as @lambda@ and also dominated by @lambda@ +-- (that is, all partial sums are less or equal): +-- +-- > dominatedPartitions lam == [ mu | mu <- partitions (weight lam), lam `dominates` mu ] +-- +dominatedPartitions :: Partition -> [Partition] +dominatedPartitions (Partition lambda) = map Partition (_dominatedPartitions lambda) + +_dominatedPartitions :: [Int] -> [[Int]] +_dominatedPartitions [] = [[]] +_dominatedPartitions lambda = go (head lambda) w dsums 0 where + + n = length lambda + w = sum lambda + dsums = scanl1 (+) (lambda ++ repeat 0) + + go _ 0 _ _ = [[]] + go !h !w (!d:ds) !e + | w > 0 = [ (a:as) | a <- [1..min h (d-e)] , as <- go a (w-a) ds (e+a) ] + | w == 0 = [[]] + | w < 0 = error "_dominatedPartitions: fatal error; shouldn't happen" + +-- | Lists all partitions of the sime weight as @mu@ and also dominating @mu@ +-- (that is, all partial sums are greater or equal): +-- +-- > dominatingPartitions mu == [ lam | lam <- partitions (weight mu), lam `dominates` mu ] +-- +dominatingPartitions :: Partition -> [Partition] +dominatingPartitions (Partition mu) = map Partition (_dominatingPartitions mu) + +_dominatingPartitions :: [Int] -> [[Int]] +_dominatingPartitions [] = [[]] +_dominatingPartitions mu = go w w dsums 0 where + + n = length mu + w = sum mu + dsums = scanl1 (+) (mu ++ repeat 0) + + go _ 0 _ _ = [[]] + go !h !w (!d:ds) !e + | w > 0 = [ (a:as) | a <- [max 0 (d-e)..min h w] , as <- go a (w-a) ds (e+a) ] + | w == 0 = [[]] + | w < 0 = error "_dominatingPartitions: fatal error; shouldn't happen" + +-------------------------------------------------------------------------------- +-- * Partitions with given number of parts + +-- | Lists partitions of @n@ into @k@ parts. +-- +-- > sort (partitionsWithKParts k n) == sort [ p | p <- partitions n , numberOfParts p == k ] +-- +-- Naive recursive algorithm. +-- +partitionsWithKParts + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = the integer we partition + -> [Partition] +partitionsWithKParts k n = map Partition $ go n k n where +{- + h = max height + k = number of parts + n = integer +-} + go !h !k !n + | k < 0 = [] + | k == 0 = if h>=0 && n==0 then [[] ] else [] + | k == 1 = if h>=n && n>=1 then [[n]] else [] + | otherwise = [ a:p | a <- [1..(min h (n-k+1))] , p <- go a (k-1) (n-a) ] + +countPartitionsWithKParts + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = the integer we partition + -> Integer +countPartitionsWithKParts k n = go n k n where + go !h !k !n + | k < 0 = 0 + | k == 0 = if h>=0 && n==0 then 1 else 0 + | k == 1 = if h>=n && n>=1 then 1 else 0 + | otherwise = sum' [ go a (k-1) (n-a) | a<-[1..(min h (n-k+1))] ] + +-------------------------------------------------------------------------------- +-- * Partitions with only odd\/distinct parts + +-- | Partitions of @n@ with only odd parts +partitionsWithOddParts :: Int -> [Partition] +partitionsWithOddParts d = map Partition (go d d) where + go _ 0 = [[]] + go !h !n = [ a:as | a<-[1,3..min n h], as <- go a (n-a) ] + +{- +-- | Partitions of @n@ with only even parts +-- +-- Note: this is not very interesting, it's just @(map.map) (2*) $ _partitions (div n 2)@ +-- +partitionsWithEvenParts :: Int -> [Partition] +partitionsWithEvenParts d = map Partition (go d d) where + go _ 0 = [[]] + go !h !n = [ a:as | a<-[2,4..min n h], as <- go a (n-a) ] +-} + +-- | Partitions of @n@ with distinct parts. +-- +-- Note: +-- +-- > length (partitionsWithDistinctParts d) == length (partitionsWithOddParts d) +-- +partitionsWithDistinctParts :: Int -> [Partition] +partitionsWithDistinctParts d = map Partition (go d d) where + go _ 0 = [[]] + go !h !n = [ a:as | a<-[1..min n h], as <- go (a-1) (n-a) ] + +-------------------------------------------------------------------------------- +-- * Sub- and super-partitions of a given partition + +-- | Returns @True@ of the first partition is a subpartition (that is, fit inside) of the second. +-- This includes equality +isSubPartitionOf :: Partition -> Partition -> Bool +isSubPartitionOf (Partition ps) (Partition qs) = and $ zipWith (<=) ps (qs ++ repeat 0) + +-- | This is provided for convenience\/completeness only, as: +-- +-- > isSuperPartitionOf q p == isSubPartitionOf p q +-- +isSuperPartitionOf :: Partition -> Partition -> Bool +isSuperPartitionOf (Partition qs) (Partition ps) = and $ zipWith (<=) ps (qs ++ repeat 0) + + +-- | Sub-partitions of a given partition with the given weight: +-- +-- > sort (subPartitions d q) == sort [ p | p <- partitions d, isSubPartitionOf p q ] +-- +subPartitions :: Int -> Partition -> [Partition] +subPartitions d (Partition ps) = map Partition (_subPartitions d ps) + +_subPartitions :: Int -> [Int] -> [[Int]] +_subPartitions d big + | null big = if d==0 then [[]] else [] + | d > sum' big = [] + | d < 0 = [] + | otherwise = go d (head big) big + where + go :: Int -> Int -> [Int] -> [[Int]] + go !k !h [] = if k==0 then [[]] else [] + go !k !h (b:bs) + | k<0 || h<0 = [] + | k==0 = [[]] + | h==0 = [] + | otherwise = [ this:rest | this <- [1..min h b] , rest <- go (k-this) this bs ] + +---------------------------------------- + +-- | All sub-partitions of a given partition +allSubPartitions :: Partition -> [Partition] +allSubPartitions (Partition ps) = map Partition (_allSubPartitions ps) + +_allSubPartitions :: [Int] -> [[Int]] +_allSubPartitions big + | null big = [[]] + | otherwise = go (head big) big + where + go _ [] = [[]] + go !h (b:bs) + | h==0 = [] + | otherwise = [] : [ this:rest | this <- [1..min h b] , rest <- go this bs ] + +---------------------------------------- + +-- | Super-partitions of a given partition with the given weight: +-- +-- > sort (superPartitions d p) == sort [ q | q <- partitions d, isSubPartitionOf p q ] +-- +superPartitions :: Int -> Partition -> [Partition] +superPartitions d (Partition ps) = map Partition (_superPartitions d ps) + +_superPartitions :: Int -> [Int] -> [[Int]] +_superPartitions dd small + | dd < w0 = [] + | null small = _partitions dd + | otherwise = go dd w1 dd (small ++ repeat 0) + where + w0 = sum' small + w1 = w0 - head small + -- d = remaining weight of the outer partition we are constructing + -- w = remaining weight of the inner partition (we need to reserve at least this amount) + -- h = max height (decreasing) + go !d !w !h (!a:as@(b:_)) + | d < 0 = [] + | d == 0 = if a == 0 then [[]] else [] + | otherwise = [ this:rest | this <- [max 1 a .. min h (d-w)] , rest <- go (d-this) (w-b) this as ] + +-------------------------------------------------------------------------------- +-- * The Pieri rule + +-- | The Pieri rule computes @s[lambda]*h[n]@ as a sum of @s[mu]@-s (each with coefficient 1). +-- +-- See for example +-- +pieriRule :: Partition -> Int -> [Partition] +pieriRule (Partition lambda) n = map Partition (_pieriRule lambda n) where + + -- | We assume here that @lambda@ is a partition (non-increasing sequence of /positive/ integers)! + _pieriRule :: [Int] -> Int -> [[Int]] + _pieriRule lambda n + | n == 0 = [lambda] + | n < 0 = [] + | otherwise = go n diffs dsums (lambda++[0]) + where + diffs = n : diffSequence lambda -- maximum we can add to a given row + dsums = reverse $ scanl1 (+) (reverse diffs) -- partial sums of remaining total we can add + go !k (d:ds) (p:ps@(q:_)) (l:ls) + | k > p = [] + | otherwise = [ h:tl | a <- [ max 0 (k-q) .. min d k ] , let h = l+a , tl <- go (k-a) ds ps ls ] + go !k [d] _ [l] = if k <= d + then if l+k>0 then [[l+k]] else [[]] + else [] + go !k [] _ _ = if k==0 then [[]] else [] + +-- | The dual Pieri rule computes @s[lambda]*e[n]@ as a sum of @s[mu]@-s (each with coefficient 1) +dualPieriRule :: Partition -> Int -> [Partition] +dualPieriRule lam n = map dualPartition $ pieriRule (dualPartition lam) n + + +{- +-- moved to "Math.Combinat.Tableaux.GelfandTsetlin" + +-- | Computes the Schur expansion of @h[n1]*h[n2]*h[n3]*...*h[nk]@ via iterating the Pieri rule +iteratedPieriRule :: Num coeff => [Int] -> Map Partition coeff +iteratedPieriRule = iteratedPieriRule' (Partition []) + +-- | Iterating the Pieri rule, we can compute the Schur expansion of +-- @h[lambda]*h[n1]*h[n2]*h[n3]*...*h[nk]@ +iteratedPieriRule' :: Num coeff => Partition -> [Int] -> Map Partition coeff +iteratedPieriRule' plambda ns = iteratedPieriRule'' (plambda,1) ns + +{-# SPECIALIZE iteratedPieriRule'' :: (Partition,Int ) -> [Int] -> Map Partition Int #-} +{-# SPECIALIZE iteratedPieriRule'' :: (Partition,Integer) -> [Int] -> Map Partition Integer #-} +iteratedPieriRule'' :: Num coeff => (Partition,coeff) -> [Int] -> Map Partition coeff +iteratedPieriRule'' (plambda,coeff0) ns = worker (Map.singleton plambda coeff0) ns where + worker old [] = old + worker old (n:ns) = worker new ns where + stuff = [ (coeff, pieriRule lam n) | (lam,coeff) <- Map.toList old ] + new = foldl' f Map.empty stuff + f t0 (c,ps) = foldl' (\t p -> Map.insertWith (+) p c t) t0 ps +-} + +-------------------------------------------------------------------------------- +-- * ASCII Ferrers diagrams + +-- | Which orientation to draw the Ferrers diagrams. +-- For example, the partition [5,4,1] corrsponds to: +-- +-- In standard English notation: +-- +-- > @@@@@ +-- > @@@@ +-- > @ +-- +-- +-- In English notation rotated by 90 degrees counter-clockwise: +-- +-- > @ +-- > @@ +-- > @@ +-- > @@ +-- > @@@ +-- +-- +-- And in French notation: +-- +-- +-- > @ +-- > @@@@ +-- > @@@@@ +-- +-- +data PartitionConvention + = EnglishNotation -- ^ English notation + | EnglishNotationCCW -- ^ English notation rotated by 90 degrees counterclockwise + | FrenchNotation -- ^ French notation (mirror of English notation to the x axis) + deriving (Eq,Show) + +-- | Synonym for @asciiFerrersDiagram\' EnglishNotation \'\@\'@ +-- +-- Try for example: +-- +-- > autoTabulate RowMajor (Right 8) (map asciiFerrersDiagram $ partitions 9) +-- +asciiFerrersDiagram :: Partition -> ASCII +asciiFerrersDiagram = asciiFerrersDiagram' EnglishNotation '@' + +asciiFerrersDiagram' :: PartitionConvention -> Char -> Partition -> ASCII +asciiFerrersDiagram' conv ch part = ASCII.asciiFromLines (map f ys) where + f n = replicate n ch + ys = case conv of + EnglishNotation -> fromPartition part + EnglishNotationCCW -> reverse $ fromPartition $ dualPartition part + FrenchNotation -> reverse $ fromPartition $ part + +instance DrawASCII Partition where + ascii = asciiFerrersDiagram + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Partitions/Multiset.hs b/Math/Combinat/Partitions/Multiset.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/Multiset.hs @@ -0,0 +1,24 @@ + +-- | Partitions of a multiset +module Math.Combinat.Partitions.Multiset where + +-------------------------------------------------------------------------------- + +import Data.Array.Unboxed +import Data.List + +import Math.Combinat.Partitions.Vector + +-------------------------------------------------------------------------------- + +-- | Partitions of a multiset. Internally, this uses the vector partition algorithm +partitionMultiset :: (Eq a, Ord a) => [a] -> [[[a]]] +partitionMultiset xs = parts where + parts = (map . map) (f . elems) temp + f ns = concat (zipWith replicate ns zs) + temp = fasc3B_algorithm_M counts + counts = map length ys + ys = group (sort xs) + zs = map head ys + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Partitions/NonCrossing.hs b/Math/Combinat/Partitions/NonCrossing.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/NonCrossing.hs @@ -0,0 +1,205 @@ + +-- | Non-crossing partitions. +-- +-- See eg. +-- +-- Non-crossing partitions of the set @[1..n]@ are encoded as lists of lists +-- in standard form: Entries decreasing in each block and blocks listed in increasing order of their first entries. +-- For example the partition in the diagram +-- +-- <> +-- +-- is represented as +-- +-- > NonCrossing [[3],[5,4,2],[7,6,1],[9,8]] +-- + +{-# LANGUAGE BangPatterns #-} +module Math.Combinat.Partitions.NonCrossing where + +-------------------------------------------------------------------------------- + +import Control.Applicative + +import Data.List +import Data.Ord + +import System.Random + +import Math.Combinat.Numbers +import Math.Combinat.LatticePaths +import Math.Combinat.Helper +import Math.Combinat.Partitions.Set +import Math.Combinat.Classes + +-------------------------------------------------------------------------------- +-- * The type of non-crossing partitions + +-- | A non-crossing partition of the set @[1..n]@ in standard form: +-- entries decreasing in each block and blocks listed in increasing order of their first entries. +newtype NonCrossing = NonCrossing [[Int]] deriving (Eq,Ord,Show,Read) + +-- | Checks whether a set partition is noncrossing. +-- +-- Implementation method: we convert to a Dyck path and then back again, and finally compare. +-- Probably not very efficient, but should be better than a naive check for crosses...) +-- +_isNonCrossing :: [[Int]] -> Bool +_isNonCrossing zzs0 = _isNonCrossingUnsafe (_standardizeNonCrossing zzs0) + +-- | Warning: This function assumes the standard ordering! +_isNonCrossingUnsafe :: [[Int]] -> Bool +_isNonCrossingUnsafe zzs = + case _nonCrossingPartitionToDyckPathMaybe zzs of + Nothing -> False + Just dyck -> case dyckPathToNonCrossingPartitionMaybe dyck of + Nothing -> False + Just (NonCrossing yys) -> yys == zzs + +-- | Convert to standard form: entries decreasing in each block +-- and blocks listed in increasing order of their first entries. +_standardizeNonCrossing :: [[Int]] -> [[Int]] +_standardizeNonCrossing = sortBy (comparing myhead) . map reverseSort where + myhead xs = case xs of + (x:xs) -> x + [] -> error "_standardizeNonCrossing: empty subset" + +fromNonCrossing :: NonCrossing -> [[Int]] +fromNonCrossing (NonCrossing xs) = xs + +toNonCrossingUnsafe :: [[Int]] -> NonCrossing +toNonCrossingUnsafe = NonCrossing + +-- | Throws an error if the input is not a non-crossing partition +toNonCrossing :: [[Int]] -> NonCrossing +toNonCrossing xxs = case toNonCrossingMaybe xxs of + Just nc -> nc + Nothing -> error "toNonCrossing: not a non-crossing partition" + +toNonCrossingMaybe :: [[Int]] -> Maybe NonCrossing +toNonCrossingMaybe xxs0 = + if _isNonCrossingUnsafe xxs + then Just $ NonCrossing xxs + else Nothing + where + xxs = _standardizeNonCrossing xxs0 + +-- | If a set partition is actually non-crossing, then we can convert it +setPartitionToNonCrossing :: SetPartition -> Maybe NonCrossing +setPartitionToNonCrossing (SetPartition zzs0) = + if _isNonCrossingUnsafe zzs + then Just $ NonCrossing zzs + else Nothing + where + zzs = _standardizeNonCrossing zzs0 + +instance HasNumberOfParts NonCrossing where + numberOfParts (NonCrossing p) = length p + +-------------------------------------------------------------------------------- +-- * Bijection to Dyck paths + +-- | Bijection between Dyck paths and noncrossing partitions +-- +-- Based on: David Callan: /Sets, Lists and Noncrossing Partitions/ +-- +-- Fails if the input is not a Dyck path. +dyckPathToNonCrossingPartition :: LatticePath -> NonCrossing +dyckPathToNonCrossingPartition = NonCrossing . go 0 [] [] [] where + go :: Int -> [Int] -> [Int] -> [[Int]] -> LatticePath -> [[Int]] + go !cnt stack small big path = + case path of + (x:xs) -> case x of + UpStep -> let cnt' = cnt + 1 in case xs of + (y:ys) -> case y of + UpStep -> go cnt' (cnt':stack) small big xs + DownStep -> go cnt' (cnt':stack) [] (reverse small : big) xs + [] -> error "dyckPathToNonCrossingPartition: last step is an UpStep (thus input was not a Dyck path)" + DownStep -> case stack of + (k:ks) -> go cnt ks (k:small) big xs + [] -> error "dyckPathToNonCrossingPartition: empty stack, shouldn't happen (thus input was not a Dyck path)" + [] -> tail $ reverse (reverse small : big) + +-- | Safe version of 'dyckPathToNonCrossingPartition' +dyckPathToNonCrossingPartitionMaybe :: LatticePath -> Maybe NonCrossing +dyckPathToNonCrossingPartitionMaybe = fmap NonCrossing . go 0 [] [] [] where + go :: Int -> [Int] -> [Int] -> [[Int]] -> LatticePath -> Maybe [[Int]] + go !cnt stack small big path = + case path of + (x:xs) -> case x of + UpStep -> let cnt' = cnt + 1 in case xs of + (y:ys) -> case y of + UpStep -> go cnt' (cnt':stack) small big xs + DownStep -> go cnt' (cnt':stack) [] (reverse small : big) xs + [] -> Nothing + DownStep -> case stack of + (k:ks) -> go cnt ks (k:small) big xs + [] -> Nothing + [] -> Just $ tail $ reverse (reverse small : big) + +-- | The inverse bijection (should never fail proper 'NonCrossing'-s) +nonCrossingPartitionToDyckPath :: NonCrossing -> LatticePath +nonCrossingPartitionToDyckPath (NonCrossing zzs) = go 0 zzs where + go !k (ys@(y:_):yys) = replicate (y-k) UpStep ++ replicate (length ys) DownStep ++ go y yys + go !k [] = [] + go _ _ = error "nonCrossingPartitionToDyckPath: shouldnt't happen" + +-- | Safe version 'nonCrossingPartitionToDyckPath' +_nonCrossingPartitionToDyckPathMaybe :: [[Int]] -> Maybe LatticePath +_nonCrossingPartitionToDyckPathMaybe = go 0 where + go !k (ys@(y:_):yys) = fmap (\zs -> replicate (y-k) UpStep ++ replicate (length ys) DownStep ++ zs) (go y yys) + go !k [] = Just [] + go _ _ = Nothing + +-------------------------------------------------------------------------------- + +{- +-- this should be mapped to NonCrossing [[3],[5,4,2],[7,6,1],[9,8]] +testpath = [u,u,u,d,u,u,d,d,d,u,u,d,d,d,u,u,d,d] where + u = UpStep + d = DownStep + +testnc = NonCrossing [[3],[5,4,2],[7,6,1],[9,8]] +-} + +-------------------------------------------------------------------------------- +-- * Generating non-crossing partitions + +-- | Lists all non-crossing partitions of @[1..n]@ +-- +-- Equivalent to (but orders of magnitude faster than) filtering out the non-crossing ones: +-- +-- > (sort $ catMaybes $ map setPartitionToNonCrossing $ setPartitions n) == sort (nonCrossingPartitions n) +-- +nonCrossingPartitions :: Int -> [NonCrossing] +nonCrossingPartitions = map dyckPathToNonCrossingPartition . dyckPaths + +-- | Lists all non-crossing partitions of @[1..n]@ into @k@ parts. +-- +-- > sort (nonCrossingPartitionsWithKParts k n) == sort [ p | p <- nonCrossingPartitions n , numberOfParts p == k ] +-- +nonCrossingPartitionsWithKParts + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = size of the set + -> [NonCrossing] +nonCrossingPartitionsWithKParts k n = map dyckPathToNonCrossingPartition $ peakingDyckPaths k n + +-- | Non-crossing partitions are counted by the Catalan numbers +countNonCrossingPartitions :: Int -> Integer +countNonCrossingPartitions = countDyckPaths + +-- | Non-crossing partitions with @k@ parts are counted by the Naranaya numbers +countNonCrossingPartitionsWithKParts + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = size of the set + -> Integer +countNonCrossingPartitionsWithKParts = countPeakingDyckPaths + +-------------------------------------------------------------------------------- + +-- | Uniformly random non-crossing partition +randomNonCrossingPartition :: RandomGen g => Int -> g -> (NonCrossing,g) +randomNonCrossingPartition n g0 = (dyckPathToNonCrossingPartition dyck, g1) where + (dyck,g1) = randomDyckPath n g0 + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Partitions/Plane.hs b/Math/Combinat/Partitions/Plane.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/Plane.hs @@ -0,0 +1,124 @@ + +-- | Plane partitions. See eg. +-- +-- Plane partitions are encoded as lists of lists of Z heights. For example the plane +-- partition in the picture +-- +-- <> +-- +-- is encoded as +-- +-- > PlanePart [ [5,4,3,3,1] +-- > , [4,4,2,1] +-- > , [3,2] +-- > , [2,1] +-- > , [1] +-- > , [1] +-- > ] +-- +{-# LANGUAGE BangPatterns #-} +module Math.Combinat.Partitions.Plane where + +-------------------------------------------------------------------------------- + +import Data.List +import Data.Array + +import Math.Combinat.Classes +import Math.Combinat.Partitions +import Math.Combinat.Tableaux as Tableaux +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * the type of plane partitions + +-- | A plane partition encoded as a tablaeu (the \"Z\" heights are the numbers) +newtype PlanePart = PlanePart [[Int]] deriving (Eq,Ord,Show) + +fromPlanePart :: PlanePart -> [[Int]] +fromPlanePart (PlanePart xs) = xs + +isValidPlanePart :: [[Int]] -> Bool +isValidPlanePart pps = + and [ table!(i,j) >= table!(i ,j+1) && + table!(i,j) >= table!(i+1,j ) + | i<-[0..y-1] , j<-[0..x-1] + ] + where + table :: Array (Int,Int) Int + table = accumArray const 0 ((0,0),(y,x)) [ ((i,j),k) | (i,ps) <- zip [0..] pps , (j,k) <- zip [0..] ps ] + y = length pps + x = maximum (map length pps) + +instance CanBeEmpty PlanePart where + isEmpty = null . fromPlanePart + empty = PlanePart [] + +-- | Throws an exception if the input is not a plane partition +toPlanePart :: [[Int]] -> PlanePart +toPlanePart pps = if isValidPlanePart pps + then PlanePart $ filter (not . null) $ map (filter (>0)) $ pps + else error "toPlanePart: not a plane partition" + +-- | The XY projected shape of a plane partition, as an integer partition +planePartShape :: PlanePart -> Partition +planePartShape = Tableaux.tableauShape . fromPlanePart + +-- | The Z height of a plane partition +planePartZHeight :: PlanePart -> Int +planePartZHeight (PlanePart xs) = + case xs of + ((h:_):_) -> h + _ -> 0 + +planePartWeight :: PlanePart -> Int +planePartWeight (PlanePart xs) = sum' (map sum' xs) + +instance HasWeight PlanePart where + weight = planePartWeight + +-------------------------------------------------------------------------------- +-- * constructing plane partitions + +singleLayer :: Partition -> PlanePart +singleLayer = PlanePart . map (\k -> replicate k 1) . fromPartition + +-- | Stacks layers of partitions into a plane partition. +-- Throws an exception if they do not form a plane partition. +stackLayers :: [Partition] -> PlanePart +stackLayers layers = if and [ isSubPartitionOf p q | (q,p) <- pairs layers ] + then unsafeStackLayers layers + else error "stackLayers: the layers do not form a plane partition" + +-- | Stacks layers of partitions into a plane partition. +-- This is unsafe in the sense that we don't check that the partitions fit on the top of each other. +unsafeStackLayers :: [Partition] -> PlanePart +unsafeStackLayers [] = PlanePart [] +unsafeStackLayers (bottom:rest) = PlanePart $ foldl addLayer (fromPlanePart $ singleLayer bottom) rest where + addLayer :: [[Int]] -> Partition -> [[Int]] + addLayer xxs (Partition ps) = [ zipWith (+) xs (replicate p 1 ++ repeat 0) | (xs,p) <- zip xxs (ps ++ repeat 0) ] + +-- | The \"layers\" of a plane partition (in direction @Z@). We should have +-- +-- > unsafeStackLayers (planePartLayers pp) == pp +-- +planePartLayers :: PlanePart -> [Partition] +planePartLayers pp@(PlanePart xs) = [ layer h | h<-[1..planePartZHeight pp] ] where + layer h = Partition $ filter (>0) $ map sum' $ (map . map) (f h) xs + f h = \k -> if k>=h then 1 else 0 + +-------------------------------------------------------------------------------- +-- * generating plane partitions + +-- | Plane partitions of a given weight +planePartitions :: Int -> [PlanePart] +planePartitions d + | d < 0 = [] + | d == 0 = [PlanePart []] + | otherwise = concat [ go (d-n) [p] | n<-[1..d] , p<-partitions n ] + where + go :: Int -> [Partition] -> [PlanePart] + go 0 acc = [unsafeStackLayers (reverse acc)] + go !rem acc@(h:_) = concat [ go (rem-k) (this:acc) | k<-[1..rem] , this <- subPartitions k h ] + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Partitions/Set.hs b/Math/Combinat/Partitions/Set.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/Set.hs @@ -0,0 +1,109 @@ + +-- | Set partitions. +-- +-- See eg. +-- + +{-# LANGUAGE BangPatterns #-} +module Math.Combinat.Partitions.Set where + +-------------------------------------------------------------------------------- + +import Data.List +import Data.Ord + +import System.Random + +import Math.Combinat.Sets +import Math.Combinat.Numbers +import Math.Combinat.Helper +import Math.Combinat.Classes +import Math.Combinat.Partitions.Integer + +-------------------------------------------------------------------------------- +-- * The type of set partitions + +-- | A partition of the set @[1..n]@ (in standard order) +newtype SetPartition = SetPartition [[Int]] deriving (Eq,Ord,Show,Read) + +_standardizeSetPartition :: [[Int]] -> [[Int]] +_standardizeSetPartition = sortBy (comparing myhead) . map sort where + myhead xs = case xs of + (x:xs) -> x + [] -> error "_standardizeSetPartition: empty subset" + +fromSetPartition :: SetPartition -> [[Int]] +fromSetPartition (SetPartition zzs) = zzs + +toSetPartitionUnsafe :: [[Int]] -> SetPartition +toSetPartitionUnsafe = SetPartition + +toSetPartition :: [[Int]] -> SetPartition +toSetPartition zzs = if _isSetPartition zzs + then SetPartition (_standardizeSetPartition zzs) + else error "toSetPartition: not a set partition" + +_isSetPartition :: [[Int]] -> Bool +_isSetPartition zzs = sort (concat zzs) == [1..n] where + n = sum' (map length zzs) + +instance HasNumberOfParts SetPartition where + numberOfParts (SetPartition p) = length p + +-------------------------------------------------------------------------------- +-- * Forgetting the set structure + +-- | The \"shape\" of a set partition is the partition we get when we forget the +-- set structure, keeping only the cardinalities. +-- +setPartitionShape :: SetPartition -> Partition +setPartitionShape (SetPartition pps) = mkPartition (map length pps) + +-------------------------------------------------------------------------------- +-- * Generating set partitions + +-- | Synonym for 'setPartitionsNaive' +setPartitions :: Int -> [SetPartition] +setPartitions = setPartitionsNaive + +-- | Synonym for 'setPartitionsWithKPartsNaive' +-- +-- > sort (setPartitionsWithKParts k n) == sort [ p | p <- setPartitions n , numberOfParts p == k ] +-- +setPartitionsWithKParts + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = size of the set + -> [SetPartition] +setPartitionsWithKParts = setPartitionsWithKPartsNaive + +-- | List all set partitions of @[1..n]@, naive algorithm +setPartitionsNaive :: Int -> [SetPartition] +setPartitionsNaive n = map (SetPartition . _standardizeSetPartition) $ go [1..n] where + go :: [Int] -> [[[Int]]] + go [] = [[]] + go (z:zs) = [ s : rest | k <- [1..n] , s0 <- choose (k-1) zs , let s = z:s0 , rest <- go (zs \\ s) ] + +-- | Set partitions of the set @[1..n]@ into @k@ parts +setPartitionsWithKPartsNaive + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = size of the set + -> [SetPartition] +setPartitionsWithKPartsNaive k n = map (SetPartition . _standardizeSetPartition) $ go k [1..n] where + go :: Int -> [Int] -> [[[Int]]] + go !k [] = if k==0 then [[]] else [] + go 1 zs = [[zs]] + go !k (z:zs) = [ s : rest | l <- [1..n] , s0 <- choose (l-1) zs , let s = z:s0 , rest <- go (k-1) (zs \\ s) ] + + +-- | Set partitions are counted by the Bell numbers +countSetPartitions :: Int -> Integer +countSetPartitions = bellNumber + +-- | Set partitions of size @k@ are counted by the Stirling numbers of second kind +countSetPartitionsWithKParts + :: Int -- ^ @k@ = number of parts + -> Int -- ^ @n@ = size of the set + -> Integer +countSetPartitionsWithKParts k n = stirling2nd n k + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Partitions/Skew.hs b/Math/Combinat/Partitions/Skew.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/Skew.hs @@ -0,0 +1,135 @@ + +-- | Skew partitions. +-- +-- Skew partitions are the difference of two integer partitions, denoted by @lambda/mu@. +-- +-- For example +-- +-- > mkSkewPartition (Partition [9,7,3,2,2,1] , Partition [5,3,2,1]) +-- +-- creates the skew partition @(9,7,3,2,2,1) / (5,3,2,1)@, which looks like +-- +-- <> +-- + +{-# LANGUAGE CPP, BangPatterns #-} +module Math.Combinat.Partitions.Skew where + +-------------------------------------------------------------------------------- + +import Data.List + +import Math.Combinat.Classes +import Math.Combinat.Partitions.Integer +import Math.Combinat.ASCII + +-------------------------------------------------------------------------------- +-- * Basics + +-- | A skew partition @lambda/mu@ is internally represented by the list @[ (mu_i , lambda_i-mu_i) | i<-[1..n] ]@ +newtype SkewPartition = SkewPartition [(Int,Int)] deriving (Eq,Ord,Show) + +-- | @mkSkewPartition (lambda,mu)@ creates the skew partition @lambda/mu@. +-- Throws an error if @mu@ is not a sub-partition of @lambda@. +mkSkewPartition :: (Partition,Partition) -> SkewPartition +mkSkewPartition ( lam@(Partition bs) , mu@(Partition as)) = if mu `isSubPartitionOf` lam + then SkewPartition $ zipWith (\b a -> (a,b-a)) bs (as ++ repeat 0) + else error "mkSkewPartition: mu should be a subpartition of lambda!" + +-- | Returns 'Nothing' if @mu@ is not a sub-partition of @lambda@. +safeSkewPartition :: (Partition,Partition) -> Maybe SkewPartition +safeSkewPartition ( lam@(Partition bs) , mu@(Partition as)) = if mu `isSubPartitionOf` lam + then Just $ SkewPartition $ zipWith (\b a -> (a,b-a)) bs (as ++ repeat 0) + else Nothing + +-- | The weight of a skew partition is the weight of the outer partition minus the +-- the weight of the inner partition (that is, the number of boxes present). +skewPartitionWeight :: SkewPartition -> Int +skewPartitionWeight (SkewPartition abs) = foldl' (+) 0 (map snd abs) + +instance HasWeight SkewPartition where + weight = skewPartitionWeight + +-- | This function \"cuts off\" the \"uninteresting parts\" of a skew partition +normalizeSkewPartition :: SkewPartition -> SkewPartition +normalizeSkewPartition (SkewPartition abs) = SkewPartition abs' where + (as,bs) = unzip abs + a0 = minimum as + k = length (takeWhile (==0) bs) + abs' = zip [ a-a0 | a <- drop k as ] (drop k bs) + +-- | Returns the outer and inner partition of a skew partition, respectively: +-- +-- > mkSkewPartition . fromSkewPartition == id +-- +fromSkewPartition :: SkewPartition -> (Partition,Partition) +fromSkewPartition (SkewPartition list) = (toPartition (zipWith (+) as bs) , toPartition (filter (>0) as)) where + (as,bs) = unzip list + +-- | The @lambda@ part of @lambda/mu@ +outerPartition :: SkewPartition -> Partition +outerPartition = fst . fromSkewPartition + +-- | The @mu@ part of @lambda/mu@ +innerPartition :: SkewPartition -> Partition +innerPartition = snd . fromSkewPartition + +-- | The dual skew partition (that is, the mirror image to the main diagonal) +dualSkewPartition :: SkewPartition -> SkewPartition +dualSkewPartition = mkSkewPartition . f . fromSkewPartition where + f (lam,mu) = (dualPartition lam, dualPartition mu) + +instance HasDuality SkewPartition where + dual = dualSkewPartition + +-------------------------------------------------------------------------------- +-- * Listing skew partitions + +-- | Lists all skew partitions with the given outer shape and given (skew) weight +skewPartitionsWithOuterShape :: Partition -> Int -> [SkewPartition] +skewPartitionsWithOuterShape outer skewWeight + | innerWeight < 0 || innerWeight > outerWeight = [] + | otherwise = [ mkSkewPartition (outer,inner) | inner <- subPartitions innerWeight outer ] + where + outerWeight = weight outer + innerWeight = outerWeight - skewWeight + +-- | Lists all skew partitions with the given outer shape and any (skew) weight +allSkewPartitionsWithOuterShape :: Partition -> [SkewPartition] +allSkewPartitionsWithOuterShape outer + = concat [ skewPartitionsWithOuterShape outer w | w<-[0..outerWeight] ] + where + outerWeight = weight outer + +-- | Lists all skew partitions with the given inner shape and given (skew) weight +skewPartitionsWithInnerShape :: Partition -> Int -> [SkewPartition] +skewPartitionsWithInnerShape inner skewWeight + | innerWeight > outerWeight = [] + | otherwise = [ mkSkewPartition (outer,inner) | outer <- superPartitions outerWeight inner ] + where + outerWeight = innerWeight + skewWeight + innerWeight = weight inner + +-------------------------------------------------------------------------------- +-- * ASCII + +asciiSkewFerrersDiagram :: SkewPartition -> ASCII +asciiSkewFerrersDiagram = asciiSkewFerrersDiagram' ('@','.') EnglishNotation + +asciiSkewFerrersDiagram' + :: (Char,Char) + -> PartitionConvention -- Orientation + -> SkewPartition + -> ASCII +asciiSkewFerrersDiagram' (outer,inner) orient (SkewPartition abs) = asciiFromLines stuff where + stuff = case orient of + EnglishNotation -> ls + EnglishNotationCCW -> reverse (transpose ls) + FrenchNotation -> reverse ls + ls = [ replicate a inner ++ replicate b outer | (a,b) <- abs ] + +instance DrawASCII SkewPartition where + ascii = asciiSkewFerrersDiagram + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Partitions/Vector.hs b/Math/Combinat/Partitions/Vector.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Partitions/Vector.hs @@ -0,0 +1,82 @@ + +-- | Vector partitions. See: +-- +-- * Donald E. Knuth: The Art of Computer Programming, vol 4, pre-fascicle 3B. +-- + +{-# LANGUAGE BangPatterns #-} +module Math.Combinat.Partitions.Vector where + +-------------------------------------------------------------------------------- + +import Data.Array.Unboxed +import Data.List + +-------------------------------------------------------------------------------- + +-- | Integer vectors. The indexing starts from 1. +type IntVector = UArray Int Int + +-- | Vector partitions. Basically a synonym for 'fasc3B_algorithm_M'. +vectorPartitions :: IntVector -> [[IntVector]] +vectorPartitions = fasc3B_algorithm_M . elems + +_vectorPartitions :: [Int] -> [[[Int]]] +_vectorPartitions = map (map elems) . fasc3B_algorithm_M + +-- | Generates all vector partitions +-- (\"algorithm M\" in Knuth). +-- The order is decreasing lexicographic. +fasc3B_algorithm_M :: [Int] -> [[IntVector]] +{- note to self: Knuth's descriptions of algorithms are still totally unreadable -} +fasc3B_algorithm_M xs = worker [start] where + + -- n = sum xs + m = length xs + + start = [ (j,x,x) | (j,x) <- zip [1..] xs ] + + worker stack@(last:_) = + case decrease stack' of + Nothing -> [visited] + Just stack'' -> visited : worker stack'' + where + stack' = subtract_rec stack + visited = map to_vector stack' + + decrease (last:rest) = + case span (\(_,_,v) -> v==0) (reverse last) of + ( _ , [(_,_,1)] ) -> case rest of + [] -> Nothing + _ -> decrease rest + ( second , (c,u,v):first ) -> Just (modified:rest) where + modified = + reverse first ++ + (c,u,v-1) : + [ (c,u,u) | (c,u,_) <- reverse second ] + _ -> error "fasc3B_algorithm_M: should not happen" + + to_vector cuvs = + accumArray (flip const) 0 (1,m) + [ (c,v) | (c,_,v) <- cuvs ] + + subtract_rec all@(last:_) = + case subtract last of + [] -> all + new -> subtract_rec (new:all) + + subtract [] = [] + subtract full@((c,u,v):rest) = + if w >= v + then (c,w,v) : subtract rest + else subtract_b full + where w = u - v + + subtract_b [] = [] + subtract_b ((c,u,v):rest) = + if w /= 0 + then (c,w,w) : subtract_b rest + else subtract_b rest + where w = u - v + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Permutations.hs b/Math/Combinat/Permutations.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Permutations.hs @@ -0,0 +1,862 @@ + +-- | Permutations. +-- +-- See eg.: +-- Donald E. Knuth: The Art of Computer Programming, vol 4, pre-fascicle 2B. +-- +-- WARNING: As of version 0.2.8.0, I changed the convention of how permutations +-- are represented internally. Also now they act on the /right/ by default! +-- + +{-# LANGUAGE CPP, BangPatterns, ScopedTypeVariables, GeneralizedNewtypeDeriving, FlexibleContexts #-} +module Math.Combinat.Permutations + ( -- * The Permutation type + Permutation (..) + , fromPermutation + , permutationArray + , permutationUArray + , uarrayToPermutationUnsafe + , isPermutation + , maybePermutation + , toPermutation + , toPermutationUnsafe + , permutationSize + -- * Disjoint cycles + , DisjointCycles (..) + , fromDisjointCycles + , disjointCyclesUnsafe + , permutationToDisjointCycles + , disjointCyclesToPermutation + , numberOfCycles + -- * Queries + , isIdentityPermutation + , isReversePermutation + , isEvenPermutation + , isOddPermutation + , signOfPermutation + , signValueOfPermutation + , module Math.Combinat.Sign -- , Sign(..) + , isCyclicPermutation + -- * Some concrete permutations + , transposition + , transpositions + , adjacentTransposition + , adjacentTranspositions + , cycleLeft + , cycleRight + , reversePermutation + -- * Inversions + , inversions + , numberOfInversions + , numberOfInversionsNaive + , numberOfInversionsMerge + , bubbleSort2 + , bubbleSort + -- * Permutation groups + , identity + , inverse + , multiply + , multiplyMany + , multiplyMany' + -- * Action of the permutation group + , permute + , permuteList + , permuteLeft , permuteRight + , permuteLeftList , permuteRightList + -- * ASCII drawing + , asciiPermutation + , asciiDisjointCycles + , twoLineNotation + , inverseTwoLineNotation + , genericTwoLineNotation + -- * List of permutations + , permutations + , _permutations + , permutationsNaive + , _permutationsNaive + , countPermutations + -- * Random permutations + , randomPermutation + , _randomPermutation + , randomCyclicPermutation + , _randomCyclicPermutation + , randomPermutationDurstenfeld + , randomCyclicPermutationSattolo + -- * Multisets + , permuteMultiset + , countPermuteMultiset + , fasc2B_algorithm_L + ) + where + +-------------------------------------------------------------------------------- + +import Control.Monad +import Control.Monad.ST + +import Data.List hiding ( permutations ) +import Data.Ord ( comparing ) + +import Data.Array (Array) +import Data.Array.ST +import Data.Array.Unboxed +import Data.Array.IArray +import Data.Array.MArray +import Data.Array.Unsafe + +import Math.Combinat.ASCII +import Math.Combinat.Classes +import Math.Combinat.Helper +import Math.Combinat.Sign +import Math.Combinat.Numbers ( factorial , binomial ) + +import System.Random + +-------------------------------------------------------------------------------- +-- * Types + +-- | A permutation. Internally it is an (unboxed) array of the integers @[1..n]@, with +-- indexing range also being @(1,n)@. +-- +-- If this array of integers is @[p1,p2,...,pn]@, then in two-line +-- notations, that represents the permutation +-- +-- > ( 1 2 3 ... n ) +-- > ( p1 p2 p3 ... pn ) +-- +-- That is, it is the permutation @sigma@ whose (right) action on the set @[1..n]@ is +-- +-- > sigma(1) = p1 +-- > sigma(2) = p2 +-- > ... +-- +-- (NOTE: this changed at version 0.2.8.0!) +-- +newtype Permutation = Permutation (UArray Int Int) deriving (Eq,Ord) -- ,Show,Read) + +instance Show Permutation where + showsPrec d (Permutation arr) + = showParen (d > 10) + $ showString "toPermutation " . showsPrec 11 (elems arr) -- app_prec = 10 + +instance Read Permutation where + readsPrec d r = readParen (d > 10) fun r where + fun r = [ (toPermutation p,t) + | ("toPermutation",s) <- lex r + , (p,t) <- readsPrec 11 s -- app_prec = 10 + ] + +instance DrawASCII Permutation where + ascii = asciiPermutation + +-- | Disjoint cycle notation for permutations. Internally it is @[[Int]]@. +-- +-- The cycles are to be understood as follows: a cycle @[c1,c2,...,ck]@ means +-- the permutation +-- +-- > ( c1 c2 c3 ... ck ) +-- > ( c2 c3 c4 ... c1 ) +-- +newtype DisjointCycles = DisjointCycles [[Int]] deriving (Eq,Ord,Show,Read) + +fromPermutation :: Permutation -> [Int] +fromPermutation (Permutation ar) = elems ar + +permutationUArray :: Permutation -> UArray Int Int +permutationUArray (Permutation ar) = ar + +-- | Note: this is slower than 'permutationUArray' +permutationArray :: Permutation -> Array Int Int +permutationArray (Permutation ar) = listArray (1,n) (elems ar) where + (1,n) = bounds ar + +-- | Assumes that the input is a permutation of the numbers @[1..n]@. +toPermutationUnsafe :: [Int] -> Permutation +toPermutationUnsafe xs = Permutation perm where + n = length xs + perm = listArray (1,n) xs + +-- | Note: Indexing starts from 1. +uarrayToPermutationUnsafe :: UArray Int Int -> Permutation +uarrayToPermutationUnsafe = Permutation + +-- | Checks whether the input is a permutation of the numbers @[1..n]@. +isPermutation :: [Int] -> Bool +isPermutation xs = (ar!0 == 0) && and [ ar!j == 1 | j<-[1..n] ] where + n = length xs + -- the zero index is an unidiomatic hack + ar = (accumArray (+) 0 (0,n) $ map f xs) :: UArray Int Int + f :: Int -> (Int,Int) + f !j = if j<1 || j>n then (0,1) else (j,1) + +-- | Checks whether the input is a permutation of the numbers @[1..n]@. +maybePermutation :: [Int] -> Maybe Permutation +maybePermutation input = runST action where + n = length input + action :: forall s. ST s (Maybe Permutation) + action = do + ar <- newArray (1,n) 0 :: ST s (STUArray s Int Int) + let go [] = return $ Just (Permutation $ listArray (1,n) input) + go (j:js) = if j<1 || j>n + then return Nothing + else do + z <- readArray ar j + writeArray ar j (z+1) + if z==0 then go js + else return Nothing + go input + +-- | Checks the input. +toPermutation :: [Int] -> Permutation +toPermutation xs = case maybePermutation xs of + Just p -> p + Nothing -> error "toPermutation: not a permutation" + +-- | Returns @n@, where the input is a permutation of the numbers @[1..n]@ +permutationSize :: Permutation -> Int +permutationSize (Permutation ar) = snd $ bounds ar + +instance HasWidth Permutation where + width = permutationSize + +-- | Checks whether the permutation is the identity permutation +isIdentityPermutation :: Permutation -> Bool +isIdentityPermutation (Permutation ar) = (elems ar == [1..n]) where + (1,n) = bounds ar + +-------------------------------------------------------------------------------- +-- * ASCII + +-- | Synonym for 'twoLineNotation' +asciiPermutation :: Permutation -> ASCII +asciiPermutation = twoLineNotation + +asciiDisjointCycles :: DisjointCycles -> ASCII +asciiDisjointCycles (DisjointCycles cycles) = final where + final = hCatWith VTop (HSepSpaces 1) boxes + boxes = [ genericTwoLineNotation (f cyc) | cyc <- cycles ] + f cyc = pairs (cyc ++ [head cyc]) + +-- | The standard two-line notation, moving the element indexed by the top row into +-- the place indexed by the corresponding element in the bottom row. +twoLineNotation :: Permutation -> ASCII +twoLineNotation (Permutation arr) = genericTwoLineNotation $ zip [1..] (elems arr) + +-- | The inverse two-line notation, where the it\'s the bottom line +-- which is in standard order. The columns of this are a permutation +-- of the columns 'twoLineNotation'. +-- +-- Remark: the top row of @inverseTwoLineNotation perm@ is the same +-- as the bottom row of @twoLineNotation (inverse perm)@. +-- +inverseTwoLineNotation :: Permutation -> ASCII +inverseTwoLineNotation (Permutation arr) = + genericTwoLineNotation $ sortBy (comparing snd) $ zip [1..] (elems arr) + +-- | Two-line notation for any set of numbers +genericTwoLineNotation :: [(Int,Int)] -> ASCII +genericTwoLineNotation xys = asciiFromLines [ topLine, botLine ] where + topLine = "( " ++ intercalate " " us ++ " )" + botLine = "( " ++ intercalate " " vs ++ " )" + pairs = [ (show x, show y) | (x,y) <- xys ] + (us,vs) = unzip (map f pairs) + f (s,t) = (s',t') where + a = length s + b = length t + c = max a b + s' = replicate (c-a) ' ' ++ s + t' = replicate (c-b) ' ' ++ t + +-------------------------------------------------------------------------------- +-- * Disjoint cycles + +fromDisjointCycles :: DisjointCycles -> [[Int]] +fromDisjointCycles (DisjointCycles cycles) = cycles + +disjointCyclesUnsafe :: [[Int]] -> DisjointCycles +disjointCyclesUnsafe = DisjointCycles + +instance DrawASCII DisjointCycles where + ascii = asciiDisjointCycles + +instance HasNumberOfCycles DisjointCycles where + numberOfCycles (DisjointCycles cycles) = length cycles + +instance HasNumberOfCycles Permutation where + numberOfCycles = numberOfCycles . permutationToDisjointCycles + +disjointCyclesToPermutation :: Int -> DisjointCycles -> Permutation +disjointCyclesToPermutation n (DisjointCycles cycles) = Permutation perm where + + pairs :: [Int] -> [(Int,Int)] + pairs xs@(x:_) = worker (xs++[x]) where + worker (x:xs@(y:_)) = (x,y):worker xs + worker _ = [] + pairs [] = error "disjointCyclesToPermutation: empty cycle" + + perm = runSTUArray $ do + ar <- newArray_ (1,n) :: ST s (STUArray s Int Int) + forM_ [1..n] $ \i -> writeArray ar i i + forM_ cycles $ \cyc -> forM_ (pairs cyc) $ \(i,j) -> writeArray ar i j + return ar -- freeze ar + +-- | Convert to disjoint cycle notation. +-- +-- This is compatible with Maple's @convert(perm,\'disjcyc\')@ +-- and also with Mathematica's @PermutationCycles[perm]@ +-- +-- Note however, that for example Mathematica uses the +-- /top row/ to represent a permutation, while we use the +-- /bottom row/ - thus even though this function looks +-- identical, the /meaning/ of both the input and output +-- is different! +-- +permutationToDisjointCycles :: Permutation -> DisjointCycles +permutationToDisjointCycles (Permutation perm) = res where + + (1,n) = bounds perm + + -- we don't want trivial cycles + f :: [Int] -> Bool + f [_] = False + f _ = True + + res = runST $ do + tag <- newArray (1,n) False + cycles <- unfoldM (step tag) 1 + return (DisjointCycles $ filter f cycles) + + step :: STUArray s Int Bool -> Int -> ST s ([Int],Maybe Int) + step tag k = do + cyc <- worker tag k k [k] + m <- next tag (k+1) + return (reverse cyc, m) + + next :: STUArray s Int Bool -> Int -> ST s (Maybe Int) + next tag k = if k > n + then return Nothing + else readArray tag k >>= \b -> if b + then next tag (k+1) + else return (Just k) + + worker :: STUArray s Int Bool -> Int -> Int -> [Int] -> ST s [Int] + worker tag k l cyc = do + writeArray tag l True + let m = perm ! l + if m == k + then return cyc + else worker tag k m (m:cyc) + +isEvenPermutation :: Permutation -> Bool +isEvenPermutation (Permutation perm) = res where + + (1,n) = bounds perm + res = runST $ do + tag <- newArray (1,n) False + cycles <- unfoldM (step tag) 1 + return $ even (sum cycles) + + step :: STUArray s Int Bool -> Int -> ST s (Int,Maybe Int) + step tag k = do + cyclen <- worker tag k k 0 + m <- next tag (k+1) + return (cyclen,m) + + next :: STUArray s Int Bool -> Int -> ST s (Maybe Int) + next tag k = if k > n + then return Nothing + else readArray tag k >>= \b -> if b + then next tag (k+1) + else return (Just k) + + worker :: STUArray s Int Bool -> Int -> Int -> Int -> ST s Int + worker tag k l cyclen = do + writeArray tag l True + let m = perm ! l + if m == k + then return cyclen + else worker tag k m (1+cyclen) + +isOddPermutation :: Permutation -> Bool +isOddPermutation = not . isEvenPermutation + +signOfPermutation :: Permutation -> Sign +signOfPermutation perm = case isEvenPermutation perm of + True -> Plus + False -> Minus + +-- | Plus 1 or minus 1. +{-# SPECIALIZE signValueOfPermutation :: Permutation -> Int #-} +{-# SPECIALIZE signValueOfPermutation :: Permutation -> Integer #-} +signValueOfPermutation :: Num a => Permutation -> a +signValueOfPermutation = signValue . signOfPermutation + +isCyclicPermutation :: Permutation -> Bool +isCyclicPermutation perm = + case cycles of + [] -> True + [cyc] -> (length cyc == n) + _ -> False + where + n = permutationSize perm + DisjointCycles cycles = permutationToDisjointCycles perm + +-------------------------------------------------------------------------------- +-- * Inversions + +-- | An /inversion/ of a permutation @sigma@ is a pair @(i,j)@ such that +-- @i sigma(j)@. +-- +-- This functions returns the inversion of a permutation. +-- +inversions :: Permutation -> [(Int,Int)] +inversions (Permutation arr) = list where + (_,n) = bounds arr + list = [ (i,j) | i<-[1..n-1], j<-[i+1..n], arr!i > arr!j ] + +-- | Returns the number of inversions: +-- +-- > numberOfInversions perm = length (inversions perm) +-- +-- Synonym for 'numberOfInversionsMerge' +-- +numberOfInversions :: Permutation -> Int +numberOfInversions = numberOfInversionsMerge + +-- | Returns the number of inversions, using the merge-sort algorithm. +-- This should be @O(n*log(n))@ +-- +numberOfInversionsMerge :: Permutation -> Int +numberOfInversionsMerge (Permutation arr) = fst (sortCnt n $ elems arr) where + (_,n) = bounds arr + + -- | First argument is length of the list. + -- Returns also the inversion count. + sortCnt :: Int -> [Int] -> (Int,[Int]) + sortCnt 0 _ = (0,[] ) + sortCnt 1 [x] = (0,[x]) + sortCnt 2 [x,y] = if x>y then (1,[y,x]) else (0,[x,y]) + sortCnt n xs = mergeCnt (sortCnt k us) (sortCnt l vs) where + k = div n 2 + l = n - k + (us,vs) = splitAt k xs + + mergeCnt :: (Int,[Int]) -> (Int,[Int]) -> (Int,[Int]) + mergeCnt (!c,us) (!d,vs) = (c+d+e,ws) where + + (e,ws) = go 0 us vs + + go !k xs [] = ( k*length xs , xs ) + go _ [] ys = ( 0 , ys) + go !k xxs@(x:xs) yys@(y:ys) = if x < y + then let (a,zs) = go k xs yys in (a+k, x:zs) + else let (a,zs) = go (k+1) xxs ys in (a , y:zs) + +-- | Returns the number of inversions, using the definition, thus it's @O(n^2)@. +-- +numberOfInversionsNaive :: Permutation -> Int +numberOfInversionsNaive (Permutation arr) = length list where + (_,n) = bounds arr + list = [ (0::Int) | i<-[1..n-1], j<-[i+1..n], arr!i > arr!j ] + +-- | Bubble sorts breaks a permutation into the product of adjacent transpositions: +-- +-- > multiplyMany' n (map (transposition n) $ bubbleSort2 perm) == perm +-- +-- Note that while this is not unique, the number of transpositions +-- equals the number of inversions. +-- +bubbleSort2 :: Permutation -> [(Int,Int)] +bubbleSort2 = map f . bubbleSort where f i = (i,i+1) + +-- | Another version of bubble sort. An entry @i@ in the return sequence means +-- the transposition @(i,i+1)@: +-- +-- > multiplyMany' n (map (adjacentTransposition n) $ bubbleSort perm) == perm +-- +bubbleSort :: Permutation -> [Int] +bubbleSort perm@(Permutation tgt) = runST action where + (_,n) = bounds tgt + + action :: forall s. ST s [Int] + action = do + fwd <- newArray_ (1,n) :: ST s (STUArray s Int Int) + inv <- newArray_ (1,n) :: ST s (STUArray s Int Int) + forM_ [1..n] $ \i -> writeArray fwd i i + forM_ [1..n] $ \i -> writeArray inv i i + + list <- forM [1..n] $ \x -> do + + let k = tgt ! x -- we take the number which will be at the @x@-th position at the end + i <- readArray inv k -- number @k@ is at the moment at position @i@ + let j = x -- but the final place is at @x@ + + let swaps = move i j + forM_ swaps $ \y -> do + + a <- readArray fwd y + b <- readArray fwd (y+1) + writeArray fwd (y+1) a + writeArray fwd y b + + u <- readArray inv a + v <- readArray inv b + writeArray inv b u + writeArray inv a v + + return swaps + + return (concat list) + + move :: Int -> Int -> [Int] + move !i !j + | j == i = [] + | j > i = [i..j-1] + | j < i = [i-1,i-2..j] + +-------------------------------------------------------------------------------- +-- * Some concrete permutations + +-- | The permutation @[n,n-1,n-2,...,2,1]@. Note that it is the inverse of itself. +reversePermutation :: Int -> Permutation +reversePermutation n = Permutation $ listArray (1,n) [n,n-1..1] + +-- | Checks whether the permutation is the reverse permutation @[n,n-1,n-2,...,2,1]. +isReversePermutation :: Permutation -> Bool +isReversePermutation (Permutation arr) = elems arr == [n,n-1..1] where (1,n) = bounds arr + +-- | A transposition (swapping two elements). +-- +-- @transposition n (i,j)@ is the permutation of size @n@ which swaps @i@\'th and @j@'th elements. +-- +transposition :: Int -> (Int,Int) -> Permutation +transposition n (i,j) = + if i>=1 && j>=1 && i<=n && j<=n + then Permutation $ listArray (1,n) [ f k | k<-[1..n] ] + else error "transposition: index out of range" + where + f k | k == i = j + | k == j = i + | otherwise = k + +-- | Product of transpositions. +-- +-- > transpositions n list == multiplyMany [ transposition n pair | pair <- list ] +-- +transpositions :: Int -> [(Int,Int)] -> Permutation +transpositions n list = Permutation (runSTUArray action) where + + action :: ST s (STUArray s Int Int) + action = do + arr <- newArray_ (1,n) + forM_ [1..n] $ \i -> writeArray arr i i + let doSwap (i,j) = do + u <- readArray arr i + v <- readArray arr j + writeArray arr i v + writeArray arr j u + mapM_ doSwap list + return arr + +-- | @adjacentTransposition n k@ swaps the elements @k@ and @(k+1)@. +adjacentTransposition :: Int -> Int -> Permutation +adjacentTransposition n k + | k>0 && k adjacentTranspositions n list == multiplyMany [ adjacentTransposition n idx | idx <- list ] +-- +adjacentTranspositions :: Int -> [Int] -> Permutation +adjacentTranspositions n list = Permutation (runSTUArray action) where + + action :: ST s (STUArray s Int Int) + action = do + arr <- newArray_ (1,n) + forM_ [1..n] $ \i -> writeArray arr i i + let doSwap i + | i<0 || i>=n = error "adjacentTranspositions: index out of range" + | otherwise = do + u <- readArray arr i + v <- readArray arr (i+1) + writeArray arr i v + writeArray arr (i+1) u + mapM_ doSwap list + return arr + +-- | The permutation which cycles a list left by one step: +-- +-- > permuteList (cycleLeft 5) "abcde" == "bcdea" +-- +-- Or in two-line notation: +-- +-- > ( 1 2 3 4 5 ) +-- > ( 2 3 4 5 1 ) +-- +cycleLeft :: Int -> Permutation +cycleLeft n = Permutation $ listArray (1,n) $ [2..n] ++ [1] + +-- | The permutation which cycles a list right by one step: +-- +-- > permuteList (cycleRight 5) "abcde" == "eabcd" +-- +-- Or in two-line notation: +-- +-- > ( 1 2 3 4 5 ) +-- > ( 5 1 2 3 4 ) +-- +cycleRight :: Int -> Permutation +cycleRight n = Permutation $ listArray (1,n) $ n : [1..n-1] + +-------------------------------------------------------------------------------- +-- * Permutation groups + +-- | Multiplies two permutations together: @p `multiply` q@ +-- means the permutation when we first apply @p@, and then @q@ +-- (that is, the natural action is the /right/ action) +-- +-- See also 'permute' for our conventions. +-- +multiply :: Permutation -> Permutation -> Permutation +multiply pi1@(Permutation perm1) pi2@(Permutation perm2) = + if (n==m) + then Permutation result + else error "multiply: permutations of different sets" + where + (_,n) = bounds perm1 + (_,m) = bounds perm2 + result = permute pi2 perm1 + +infixr 7 `multiply` + +-- | The inverse permutation. +inverse :: Permutation -> Permutation +inverse (Permutation perm1) = Permutation result + where + result = array (1,n) $ map swap $ assocs perm1 + (_,n) = bounds perm1 + +-- | The identity (or trivial) permutation. +identity :: Int -> Permutation +identity n = Permutation $ listArray (1,n) [1..n] + +-- | Multiply together a /non-empty/ list of permutations (the reason for requiring the list to +-- be non-empty is that we don\'t know the size of the result). See also 'multiplyMany''. +multiplyMany :: [Permutation] -> Permutation +multiplyMany [] = error "multiplyMany: empty list, we don't know size of the result" +multiplyMany ps = foldl1' multiply ps + +-- | Multiply together a (possibly empty) list of permutations, all of which has size @n@ +multiplyMany' :: Int -> [Permutation] -> Permutation +multiplyMany' n [] = identity n +multiplyMany' n ps@(p:_) = if n == permutationSize p + then foldl1' multiply ps + else error "multiplyMany': incompatible permutation size(s)" + +-------------------------------------------------------------------------------- +-- * Action of the permutation group + +-- | /Right/ action of a permutation on a set. If our permutation is +-- encoded with the sequence @[p1,p2,...,pn]@, then in the +-- two-line notation we have +-- +-- > ( 1 2 3 ... n ) +-- > ( p1 p2 p3 ... pn ) +-- +-- We adopt the convention that permutations act /on the right/ +-- (as in Knuth): +-- +-- > permute pi2 (permute pi1 set) == permute (pi1 `multiply` pi2) set +-- +-- Synonym to 'permuteRight' +-- +{-# SPECIALIZE permute :: Permutation -> Array Int b -> Array Int b #-} +{-# SPECIALIZE permute :: Permutation -> UArray Int Int -> UArray Int Int #-} +permute :: IArray arr b => Permutation -> arr Int b -> arr Int b +permute = permuteRight + +-- | Right action on lists. Synonym to 'permuteListRight' +-- +permuteList :: Permutation -> [a] -> [a] +permuteList = permuteRightList + +-- | The right (standard) action of permutations on sets. +-- +-- > permuteRight pi2 (permuteRight pi1 set) == permuteRight (pi1 `multiply` pi2) set +-- +-- The second argument should be an array with bounds @(1,n)@. +-- The function checks the array bounds. +-- +{-# SPECIALIZE permuteRight :: Permutation -> Array Int b -> Array Int b #-} +{-# SPECIALIZE permuteRight :: Permutation -> UArray Int Int -> UArray Int Int #-} +permuteRight :: IArray arr b => Permutation -> arr Int b -> arr Int b +permuteRight pi@(Permutation perm) ar = + if (a==1) && (b==n) + then listArray (1,n) [ ar!(perm!i) | i <- [1..n] ] + else error "permuteRight: array bounds do not match" + where + (_,n) = bounds perm + (a,b) = bounds ar + +-- | The right (standard) action on a list. The list should be of length @n@. +-- +-- > fromPermutation perm == permuteRightList perm [1..n] +-- +permuteRightList :: forall a . Permutation -> [a] -> [a] +permuteRightList perm xs = elems $ permuteRight perm $ arr where + arr = listArray (1,n) xs :: Array Int a + n = permutationSize perm + +-- | The left (opposite) action of the permutation group. +-- +-- > permuteLeft pi2 (permuteLeft pi1 set) == permuteLeft (pi2 `multiply` pi1) set +-- +-- It is related to 'permuteLeft' via: +-- +-- > permuteLeft pi arr == permuteRight (inverse pi) arr +-- > permuteRight pi arr == permuteLeft (inverse pi) arr +-- +{-# SPECIALIZE permuteLeft :: Permutation -> Array Int b -> Array Int b #-} +{-# SPECIALIZE permuteLeft :: Permutation -> UArray Int Int -> UArray Int Int #-} +permuteLeft :: IArray arr b => Permutation -> arr Int b -> arr Int b +permuteLeft pi@(Permutation perm) ar = + -- permuteRight (inverse pi) ar + if (a==1) && (b==n) + then array (1,n) [ ( perm!i , ar!i ) | i <- [1..n] ] + else error "permuteLeft: array bounds do not match" + where + (_,n) = bounds perm + (a,b) = bounds ar + +-- | The left (opposite) action on a list. The list should be of length @n@. +-- +-- > permuteLeftList perm set == permuteList (inverse perm) set +-- > fromPermutation (inverse perm) == permuteLeftList perm [1..n] +-- +permuteLeftList :: forall a. Permutation -> [a] -> [a] +permuteLeftList perm xs = elems $ permuteLeft perm $ arr where + arr = listArray (1,n) xs :: Array Int a + n = permutationSize perm + +-------------------------------------------------------------------------------- +-- * Permutations of distinct elements + +-- | A synonym for 'permutationsNaive' +permutations :: Int -> [Permutation] +permutations = permutationsNaive + +_permutations :: Int -> [[Int]] +_permutations = _permutationsNaive + +-- | All permutations of @[1..n]@ in lexicographic order, naive algorithm. +permutationsNaive :: Int -> [Permutation] +permutationsNaive n = map toPermutationUnsafe $ _permutations n + +_permutationsNaive :: Int -> [[Int]] +_permutationsNaive 0 = [[]] +_permutationsNaive 1 = [[1]] +_permutationsNaive n = helper [1..n] where + helper [] = [[]] + helper xs = [ i : ys | i <- xs , ys <- helper (xs `minus` i) ] + minus [] _ = [] + minus (x:xs) i = if x < i then x : minus xs i else xs + +-- | # = n! +countPermutations :: Int -> Integer +countPermutations = factorial + +-------------------------------------------------------------------------------- +-- * Random permutations + +-- | A synonym for 'randomPermutationDurstenfeld'. +randomPermutation :: RandomGen g => Int -> g -> (Permutation,g) +randomPermutation = randomPermutationDurstenfeld + +_randomPermutation :: RandomGen g => Int -> g -> ([Int],g) +_randomPermutation n rndgen = (fromPermutation perm, rndgen') where + (perm, rndgen') = randomPermutationDurstenfeld n rndgen + +-- | A synonym for 'randomCyclicPermutationSattolo'. +randomCyclicPermutation :: RandomGen g => Int -> g -> (Permutation,g) +randomCyclicPermutation = randomCyclicPermutationSattolo + +_randomCyclicPermutation :: RandomGen g => Int -> g -> ([Int],g) +_randomCyclicPermutation n rndgen = (fromPermutation perm, rndgen') where + (perm, rndgen') = randomCyclicPermutationSattolo n rndgen + +-- | Generates a uniformly random permutation of @[1..n]@. +-- Durstenfeld's algorithm (see ). +randomPermutationDurstenfeld :: RandomGen g => Int -> g -> (Permutation,g) +randomPermutationDurstenfeld = randomPermutationDurstenfeldSattolo False + +-- | Generates a uniformly random /cyclic/ permutation of @[1..n]@. +-- Sattolo's algorithm (see ). +randomCyclicPermutationSattolo :: RandomGen g => Int -> g -> (Permutation,g) +randomCyclicPermutationSattolo = randomPermutationDurstenfeldSattolo True + +randomPermutationDurstenfeldSattolo :: RandomGen g => Bool -> Int -> g -> (Permutation,g) +randomPermutationDurstenfeldSattolo isSattolo n rnd = res where + res = runST $ do + ar <- newArray_ (1,n) + forM_ [1..n] $ \i -> writeArray ar i i + rnd' <- worker n (if isSattolo then n-1 else n) rnd ar + perm <- Data.Array.Unsafe.unsafeFreeze ar + return (Permutation perm, rnd') + worker :: RandomGen g => Int -> Int -> g -> STUArray s Int Int -> ST s g + worker n m rnd ar = + if n==1 + then return rnd + else do + let (k,rnd') = randomR (1,m) rnd + when (k /= n) $ do + y <- readArray ar k + z <- readArray ar n + writeArray ar n y + writeArray ar k z + worker (n-1) (m-1) rnd' ar + +-------------------------------------------------------------------------------- +-- * Permutations of a multiset + +-- | Generates all permutations of a multiset. +-- The order is lexicographic. A synonym for 'fasc2B_algorithm_L' +permuteMultiset :: (Eq a, Ord a) => [a] -> [[a]] +permuteMultiset = fasc2B_algorithm_L + +-- | # = \\frac { (\sum_i n_i) ! } { \\prod_i (n_i !) } +countPermuteMultiset :: (Eq a, Ord a) => [a] -> Integer +countPermuteMultiset xs = factorial n `div` product [ factorial (length z) | z <- group ys ] + where + ys = sort xs + n = length xs + +-- | Generates all permutations of a multiset +-- (based on \"algorithm L\" in Knuth; somewhat less efficient). +-- The order is lexicographic. +fasc2B_algorithm_L :: (Eq a, Ord a) => [a] -> [[a]] +fasc2B_algorithm_L xs = unfold1 next (sort xs) where + + -- next :: [a] -> Maybe [a] + next xs = case findj (reverse xs,[]) of + Nothing -> Nothing + Just ( (l:ls) , rs) -> Just $ inc l ls (reverse rs,[]) + Just ( [] , _ ) -> error "permute: should not happen" + + -- we use simple list zippers: (left,right) + -- findj :: ([a],[a]) -> Maybe ([a],[a]) + findj ( xxs@(x:xs) , yys@(y:_) ) = if x >= y + then findj ( xs , x : yys ) + else Just ( xxs , yys ) + findj ( x:xs , [] ) = findj ( xs , [x] ) + findj ( [] , _ ) = Nothing + + -- inc :: a -> [a] -> ([a],[a]) -> [a] + inc !u us ( (x:xs) , yys ) = if u >= x + then inc u us ( xs , x : yys ) + else reverse (x:us) ++ reverse (u:yys) ++ xs + inc _ _ ( [] , _ ) = error "permute: should not happen" + +-------------------------------------------------------------------------------- + + diff --git a/Math/Combinat/Sets.hs b/Math/Combinat/Sets.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Sets.hs @@ -0,0 +1,212 @@ + +-- | Subsets. + +{-# LANGUAGE BangPatterns, Rank2Types #-} +module Math.Combinat.Sets + ( + -- * Choices + choose_ , choose , choose' , choose'' , chooseTagged + -- * Compositions + , combine , compose + -- * Tensor products + , tuplesFromList + , listTensor + -- * Sublists + , kSublists + , sublists + , countKSublists + , countSublists + -- * Random choice + , randomChoice + ) + where + +-------------------------------------------------------------------------------- + +import Data.List ( sort , mapAccumL ) +import System.Random + +import Control.Monad +import Control.Monad.ST +import Data.Array.ST +import Data.Array.MArray + +-- import Data.Map (Map) +-- import qualified Data.Map as Map + +import Math.Combinat.Numbers ( binomial ) +import Math.Combinat.Helper ( swap ) + +-------------------------------------------------------------------------------- +-- * choices + + +-- | @choose_ k n@ returns all possible ways of choosing @k@ disjoint elements from @[1..n]@ +-- +-- > choose_ k n == choose k [1..n] +-- +choose_ :: Int -> Int -> [[Int]] +choose_ k n = if n<0 || k<0 + then error "choose_: n and k should nonnegative" + else if k>n || k<0 + then [] + else choose k [1..n] + +-- | All possible ways to choose @k@ elements from a list, without +-- repetitions. \"Antisymmetric power\" for lists. Synonym for 'kSublists'. +choose :: Int -> [a] -> [[a]] +choose 0 _ = [[]] +choose k [] = [] +choose k (x:xs) = map (x:) (choose (k-1) xs) ++ choose k xs + +-- | A version of 'choose' which also returns the complementer sets. +-- +-- > choose k = map fst . choose' k +-- +choose' :: Int -> [a] -> [([a],[a])] +choose' 0 xs = [([],xs)] +choose' k [] = [] +choose' k (x:xs) = map f (choose' (k-1) xs) ++ map g (choose' k xs) where + f (as,bs) = (x:as , bs) + g (as,bs) = ( as , x:bs) + +-- | Another variation of 'choose''. This satisfies +-- +-- > choose'' k == map (\(xs,ys) -> (map fst xs, map snd ys)) . choose' k +-- +choose'' :: Int -> [(a,b)] -> [([a],[b])] +choose'' 0 xys = [([] , map snd xys)] +choose'' k [] = [] +choose'' k ((x,y):xs) = map f (choose'' (k-1) xs) ++ map g (choose'' k xs) where + f (as,bs) = (x:as , bs) + g (as,bs) = ( as , y:bs) + +-- | Another variation on 'choose' which tags the elements based on whether they are part of +-- the selected subset ('Right') or not ('Left'): +-- +-- > choose k = map rights . chooseTagged k +-- +chooseTagged :: Int -> [a] -> [[Either a a]] +chooseTagged 0 xs = [map Left xs] +chooseTagged k [] = [] +chooseTagged k (x:xs) = map f (chooseTagged (k-1) xs) ++ map g (chooseTagged k xs) where + f eis = Right x : eis + g eis = Left x : eis + +-- | All possible ways to choose @k@ elements from a list, /with repetitions/. +-- \"Symmetric power\" for lists. See also "Math.Combinat.Compositions". +-- TODO: better name? +combine :: Int -> [a] -> [[a]] +combine 0 _ = [[]] +combine k [] = [] +combine k xxs@(x:xs) = map (x:) (combine (k-1) xxs) ++ combine k xs + +-- | A synonym for 'combine'. +compose :: Int -> [a] -> [[a]] +compose = combine + +-------------------------------------------------------------------------------- +-- * tensor products + +-- | \"Tensor power\" for lists. Special case of 'listTensor': +-- +-- > tuplesFromList k xs == listTensor (replicate k xs) +-- +-- See also "Math.Combinat.Tuples". +-- TODO: better name? +tuplesFromList :: Int -> [a] -> [[a]] +tuplesFromList 0 _ = [[]] +tuplesFromList k xs = [ (y:ys) | ys <- tuplesFromList (k-1) xs , y <- xs ] +--the order seems to be very important, the wrong order causes a memory leak! +--tuplesFromList k xs = [ (y:ys) | y <- xs, ys <- tuplesFromList (k-1) xs ] + +-- | \"Tensor product\" for lists. +listTensor :: [[a]] -> [[a]] +listTensor [] = [[]] +listTensor (xs:xss) = [ y:ys | ys <- listTensor xss , y <- xs ] +--the order seems to be very important, the wrong order causes a memory leak! +--listTensor (xs:xss) = [ y:ys | y <- xs, ys <- listTensor xss ] + +-------------------------------------------------------------------------------- +-- * sublists + +-- | Sublists of a list having given number of elements. Synonym for 'choose'. +kSublists :: Int -> [a] -> [[a]] +kSublists = choose + +-- | @# = \binom { n } { k }@. +countKSublists :: Int -> Int -> Integer +countKSublists k n = binomial n k + +-- | All sublists of a list. +sublists :: [a] -> [[a]] +sublists [] = [[]] +sublists (x:xs) = sublists xs ++ map (x:) (sublists xs) +--the order seems to be very important, the wrong order causes a memory leak! +--sublists (x:xs) = map (x:) (sublists xs) ++ sublists xs + +-- | @# = 2^n@. +countSublists :: Int -> Integer +countSublists n = 2 ^ n + +-------------------------------------------------------------------------------- +-- * random choice + +-- | @randomChoice k n@ returns a uniformly random choice of @k@ elements from the set @[1..n]@ +-- +-- Example: +-- +-- > do +-- > cs <- replicateM 10000 (getStdRandom (randomChoice 3 7)) +-- > mapM_ print $ histogram cs +-- +randomChoice :: RandomGen g => Int -> Int -> g -> ([Int],g) +randomChoice k n g0 = + if k>n || k<0 + then error "randomChoice: k out of range" + else (makeChoiceFromIndicesKnuth n as, g1) + where + -- choose numbers from [1..n], [1..n-1], [1..n-2] etc + (g1,as) = mapAccumL (\g m -> swap (randomR (1,m) g)) g0 [n,n-1..n-k+1] + +-------------------------------------------------------------------------------- + +-- | From a list of $k$ numbers, where the first is in the interval @[1..n]@, +-- the second in @[1..n-1]@, the third in @[1..n-2]@, we create a size @k@ subset of @n@. +-- +-- Knuth's method. The first argument is the number @n@. +-- +makeChoiceFromIndicesKnuth :: Int -> [Int] -> [Int] +makeChoiceFromIndicesKnuth n xs = + runST $ do + arr <- newArray_ (1,n) :: forall s. ST s (STUArray s Int Int) + forM_ [1..n] $ \(!j) -> writeArray arr j j + forM_ (zip [n,n-1..] xs) $ \(!j,!i) -> do + a <- readArray arr j + b <- readArray arr i + writeArray arr j b + writeArray arr i a + sel <- forM (zip [n,n-1..] xs) $ \(!j,_) -> readArray arr j + return (sort sel) + +-- | From a list of $k$ numbers, where the first is in the interval @[1..n]@, +-- the second in @[1..n-1]@, the third in @[1..n-2]@, we create a size @k@ subset of @n@. +makeChoiceFromIndicesNaive :: [Int] -> [Int] +makeChoiceFromIndicesNaive = sort . go [] where + + go :: [Int] -> [Int] -> [Int] + go acc (b:bs) = b' : go (insert b' acc) bs where b' = skip b acc + go _ [] = [] + + -- skip over the already occupied positions. Second argument should be a sorted list + skip :: Int -> [Int] -> Int + skip x (y:ys) = if x>=y then skip (x+1) ys else x + skip x [] = x + + -- insert into a sorted list + insert :: Int -> [Int] -> [Int] + insert x (y:ys) = if x<=y then x:y:ys else y : insert x ys + insert x [] = [x] + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Sign.hs b/Math/Combinat/Sign.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Sign.hs @@ -0,0 +1,90 @@ + +-- | Signs + +{-# LANGUAGE BangPatterns #-} +module Math.Combinat.Sign where + +-------------------------------------------------------------------------------- + +import Data.Monoid +import System.Random + +-------------------------------------------------------------------------------- + +data Sign + = Plus -- hmm, this way @Plus < Minus@, not sure about that + | Minus + deriving (Eq,Ord,Show,Read) + +instance Semigroup Sign where + (<>) = mulSign + +instance Monoid Sign where + mempty = Plus + mconcat = productOfSigns + +instance Random Sign where + random g = let (b,g') = random g in (if b then Plus else Minus, g') + randomR (u,v) g = let (y,g') = random g in (if u==v then u else y , g') + +isPlus, isMinus :: Sign -> Bool +isPlus s = case s of { Plus -> True ; _ -> False } +isMinus s = case s of { Minus -> True ; _ -> False } + +{-# SPECIALIZE signValue :: Sign -> Int #-} +{-# SPECIALIZE signValue :: Sign -> Integer #-} + +-- | @+1@ or @-1@ +signValue :: Num a => Sign -> a +signValue s = case s of + Plus -> 1 + Minus -> -1 + +{-# SPECIALIZE signed :: Sign -> Int -> Int #-} +{-# SPECIALIZE signed :: Sign -> Integer -> Integer #-} + +-- | Negate the second argument if the first is 'Minus' +signed :: Num a => Sign -> a -> a +signed s y = case s of + Plus -> y + Minus -> negate y + +{-# SPECIALIZE paritySign :: Int -> Sign #-} +{-# SPECIALIZE paritySign :: Integer -> Sign #-} + +-- | 'Plus' if even, 'Minus' if odd +paritySign :: Integral a => a -> Sign +paritySign x = if even x then Plus else Minus + +{-# SPECIALIZE paritySignValue :: Int -> Integer #-} +{-# SPECIALIZE paritySignValue :: Integer -> Integer #-} + +-- | @(-1)^k@ +paritySignValue :: Integral a => a -> Integer +paritySignValue k = if odd k then (-1) else 1 + +{-# SPECIALIZE negateIfOdd :: Int -> Int -> Int #-} +{-# SPECIALIZE negateIfOdd :: Int -> Integer -> Integer #-} + +-- | Negate the second argument if the first is odd +negateIfOdd :: (Integral a, Num b) => a -> b -> b +negateIfOdd k y = if even k then y else negate y + +oppositeSign :: Sign -> Sign +oppositeSign s = case s of + Plus -> Minus + Minus -> Plus + +mulSign :: Sign -> Sign -> Sign +mulSign s1 s2 = case s1 of + Plus -> s2 + Minus -> oppositeSign s2 + +productOfSigns :: [Sign] -> Sign +productOfSigns = go Plus where + go !acc [] = acc + go !acc (x:xs) = case x of + Plus -> go acc xs + Minus -> go (oppositeSign acc) xs + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Tableaux.hs b/Math/Combinat/Tableaux.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Tableaux.hs @@ -0,0 +1,241 @@ + +-- | Young tableaux and similar gadgets. +-- +-- See e.g. William Fulton: Young Tableaux, with Applications to +-- Representation theory and Geometry (CUP 1997). +-- +-- The convention is that we use +-- the English notation, and we store the tableaux as lists of the rows. +-- +-- That is, the following standard Young tableau of shape [5,4,1] +-- +-- > 1 3 4 6 7 +-- > 2 5 8 10 +-- > 9 +-- +-- <> +-- +-- is encoded conveniently as +-- +-- > [ [ 1 , 3 , 4 , 6 , 7 ] +-- > , [ 2 , 5 , 8 ,10 ] +-- > , [ 9 ] +-- > ] +-- + +{-# LANGUAGE CPP, BangPatterns, FlexibleInstances, TypeSynonymInstances, MultiParamTypeClasses #-} +module Math.Combinat.Tableaux where + +-------------------------------------------------------------------------------- + +import Data.List + +import Math.Combinat.Classes +import Math.Combinat.Numbers (factorial,binomial) +import Math.Combinat.Partitions +import Math.Combinat.ASCII +import Math.Combinat.Helper + +import Data.Map.Strict (Map) +import qualified Data.Map.Strict as Map + +-------------------------------------------------------------------------------- +-- * Basic stuff + +-- | A tableau is simply represented as a list of lists. +type Tableau a = [[a]] + +-- | ASCII diagram of a tableau +asciiTableau :: Show a => Tableau a -> ASCII +asciiTableau t = tabulate (HRight,VTop) (HSepSpaces 1, VSepEmpty) + $ (map . map) asciiShow + $ t + +instance CanBeEmpty (Tableau a) where + empty = [] + isEmpty = null + +instance Show a => DrawASCII (Tableau a) where + ascii = asciiTableau + +_tableauShape :: Tableau a -> [Int] +_tableauShape t = map length t + +-- | The shape of a tableau +tableauShape :: Tableau a -> Partition +tableauShape t = toPartition (_tableauShape t) + +instance HasShape (Tableau a) Partition where + shape = tableauShape + +-- | Number of entries +tableauWeight :: Tableau a -> Int +tableauWeight = sum' . map length + +instance HasWeight (Tableau a) where + weight = tableauWeight + +-- | The dual of the tableau is the mirror image to the main diagonal. +dualTableau :: Tableau a -> Tableau a +dualTableau = transpose + +instance HasDuality (Tableau a) where + dual = dualTableau + +-- | The content of a tableau is the list of its entries. The ordering is from the left to the right and +-- then from the top to the bottom +tableauContent :: Tableau a -> [a] +tableauContent = concat + +-- | An element @(i,j)@ of the resulting tableau (which has shape of the +-- given partition) means that the vertical part of the hook has length @i@, +-- and the horizontal part @j@. The /hook length/ is thus @i+j-1@. +-- +-- Example: +-- +-- > > mapM_ print $ hooks $ toPartition [5,4,1] +-- > [(3,5),(2,4),(2,3),(2,2),(1,1)] +-- > [(2,4),(1,3),(1,2),(1,1)] +-- > [(1,1)] +-- +hooks :: Partition -> Tableau (Int,Int) +hooks part = zipWith f p [1..] where + p = fromPartition part + q = _dualPartition p + f l i = zipWith (\x y -> (x-i+1,y)) q [l,l-1..1] + +hookLengths :: Partition -> Tableau Int +hookLengths part = (map . map) (\(i,j) -> i+j-1) (hooks part) + +-------------------------------------------------------------------------------- +-- * Row and column words + +-- | The /row word/ of a tableau is the list of its entry read from the right to the left and then +-- from the top to the bottom. +rowWord :: Tableau a -> [a] +rowWord = concat . reverse + +-- | /Semistandard/ tableaux can be reconstructed from their row words +rowWordToTableau :: Ord a => [a] -> Tableau a +rowWordToTableau xs = reverse rows where + rows = break xs + break [] = [[]] + break [x] = [[x]] + break (x:xs@(y:_)) = if x>y + then [x] : break xs + else let (h:t) = break xs in (x:h):t + +-- | The /column word/ of a tableau is the list of its entry read from the bottom to the top and then from the left to the right +columnWord :: Tableau a -> [a] +columnWord = rowWord . transpose + +-- | /Standard/ tableaux can be reconstructed from either their column or row words +columnWordToTableau :: Ord a => [a] -> Tableau a +columnWordToTableau = transpose . rowWordToTableau + +-- | Checks whether a sequence of positive integers is a /lattice word/, +-- which means that in every initial part of the sequence any number @i@ +-- occurs at least as often as the number @i+1@ +-- +isLatticeWord :: [Int] -> Bool +isLatticeWord = go Map.empty where + go :: Map Int Int -> [Int] -> Bool + go _ [] = True + go !table (i:is) = + if check i + then go table' is + else False + where + table' = Map.insertWith (+) i 1 table + check j = j==1 || cnt (j-1) >= cnt j + cnt j = case Map.lookup j table' of + Just k -> k + Nothing -> 0 + +-------------------------------------------------------------------------------- +-- * Semistandard Young tableaux + +-- | A tableau is /semistandard/ if its entries are weekly increasing horizontally +-- and strictly increasing vertically +isSemiStandardTableau :: Tableau Int -> Bool +isSemiStandardTableau t = weak && strict where + weak = and [ isWeaklyIncreasing xs | xs <- t ] + strict = and [ isStrictlyIncreasing ys | ys <- dt ] + dt = dualTableau t + +-- | Semistandard Young tableaux of given shape, \"naive\" algorithm +semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int] +semiStandardYoungTableaux n part = worker (repeat 0) shape where + shape = fromPartition part + worker _ [] = [[]] + worker prevRow (s:ss) + = [ (r:rs) | r <- row n s 1 prevRow, rs <- worker (map (+1) r) ss ] + + -- weekly increasing lists of length @len@, pointwise at least @xs@, + -- maximum value @n@, minimum value @prev@. + row :: Int -> Int -> Int -> [Int] -> [[Int]] + row _ 0 _ _ = [[]] + row n len prev (x:xs) = [ (a:as) | a <- [max x prev..n] , as <- row n (len-1) a xs ] + +-- | Stanley's hook formula (cf. Fulton page 55) +countSemiStandardYoungTableaux :: Int -> Partition -> Integer +countSemiStandardYoungTableaux n shape = k `div` h where + h = product $ map fromIntegral $ concat $ hookLengths shape + k = product [ fromIntegral (n+j-i) | (i,j) <- elements shape ] + + +-------------------------------------------------------------------------------- +-- * Standard Young tableaux + +-- | A tableau is /standard/ if it is semistandard and its content is exactly @[1..n]@, +-- where @n@ is the weight. +isStandardTableau :: Tableau Int -> Bool +isStandardTableau t = isSemiStandardTableau t && sort (concat t) == [1..n] where + n = sum [ length xs | xs <- t ] + +-- | Standard Young tableaux of a given shape. +-- Adapted from John Stembridge, +-- . +standardYoungTableaux :: Partition -> [Tableau Int] +standardYoungTableaux shape' = map rev $ tableaux shape where + shape = fromPartition shape' + rev = reverse . map reverse + tableaux :: [Int] -> [Tableau Int] + tableaux p = + case p of + [] -> [[]] + [n] -> [[[n,n-1..1]]] + _ -> worker (n,k) 0 [] p + where + n = sum p + k = length p + worker :: (Int,Int) -> Int -> [Int] -> [Int] -> [Tableau Int] + worker _ _ _ [] = [] + worker nk i ls (x:rs) = case rs of + (y:_) -> if x==y + then worker nk (i+1) (x:ls) rs + else worker2 nk i ls x rs + [] -> worker2 nk i ls x rs + worker2 :: (Int,Int) -> Int -> [Int] -> Int -> [Int] -> [Tableau Int] + worker2 nk@(n,k) i ls x rs = new ++ worker nk (i+1) (x:ls) rs where + old = if x>1 + then tableaux $ reverse ls ++ (x-1) : rs + else map ([]:) $ tableaux $ reverse ls ++ rs + a = k-1-i + new = {- debug ( i , a , head old , f a (head old) ) $ -} + map (f a) old + f :: Int -> Tableau Int -> Tableau Int + f _ [] = [] + f 0 (t:ts) = (n:t) : f (-1) ts + f j (t:ts) = t : f (j-1) ts + +-- | hook-length formula +countStandardYoungTableaux :: Partition -> Integer +countStandardYoungTableaux part = {- debug (hookLengths part) $ -} + factorial n `div` h where + h = product $ map fromIntegral $ concat $ hookLengths part + n = weight part + +-------------------------------------------------------------------------------- + + diff --git a/Math/Combinat/Tableaux/GelfandTsetlin.hs b/Math/Combinat/Tableaux/GelfandTsetlin.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Tableaux/GelfandTsetlin.hs @@ -0,0 +1,341 @@ + +-- | Gelfand-Tsetlin patterns and Kostka numbers. +-- +-- Gelfand-Tsetlin patterns (or tableaux) are triangular arrays like +-- +-- > [ 3 ] +-- > [ 3 , 2 ] +-- > [ 3 , 1 , 0 ] +-- > [ 2 , 0 , 0 , 0 ] +-- +-- with both rows and columns non-increasing non-negative integers. +-- Note: these are in bijection with the semi-standard Young tableaux. +-- +-- If we add the further restriction that +-- the top diagonal reads @lambda@, +-- and the diagonal sums are partial sums of @mu@, where @lambda@ and @mu@ are two +-- partitions (in this case @lambda=[3,2]@ and @mu=[2,1,1,1]@), +-- then the number of the resulting patterns +-- or tableaux is the Kostka number @K(lambda,mu)@. +-- Actually @mu@ doesn't even need to the be non-increasing. +-- + +{-# LANGUAGE BangPatterns, ScopedTypeVariables #-} +module Math.Combinat.Tableaux.GelfandTsetlin where + +-------------------------------------------------------------------------------- + +import Data.List +import Data.Maybe +import Data.Monoid +import Data.Ord + +import Control.Monad +import Control.Monad.Trans.State + +import Data.Map (Map) +import qualified Data.Map as Map + +import Math.Combinat.Partitions.Integer +import Math.Combinat.Tableaux +import Math.Combinat.Helper +import Math.Combinat.ASCII + +-------------------------------------------------------------------------------- +-- * Kostka numbers + +-- | Kostka numbers (via counting Gelfand-Tsetlin patterns). See for example +-- +-- @K(lambda,mu)==0@ unless @lambda@ dominates @mu@: +-- +-- > [ mu | mu <- partitions (weight lam) , kostkaNumber lam mu > 0 ] == dominatedPartitions lam +-- +kostkaNumber :: Partition -> Partition -> Int +kostkaNumber = countKostkaGelfandTsetlinPatterns + +-- | Very naive (and slow) implementation of Kostka numbers, for reference. +kostkaNumberReferenceNaive :: Partition -> Partition -> Int +kostkaNumberReferenceNaive plambda pmu@(Partition mu) = length stuff where + stuff = [ (1::Int) | t <- semiStandardYoungTableaux k plambda , cond t ] + k = length mu + cond t = [ (head xs, length xs) | xs <- group (sort $ concat t) ] == zip [1..] mu + +-------------------------------------------------------------------------------- + +-- | Lists all (positive) Kostka numbers @K(lambda,mu)@ with the given @lambda@: +-- +-- > kostkaNumbersWithGivenLambda lambda == Map.fromList [ (mu , kostkaNumber lambda mu) | mu <- dominatedPartitions lambda ] +-- +-- It's much faster than computing the individual Kostka numbers, but not as fast +-- as it could be. +-- +{-# SPECIALIZE kostkaNumbersWithGivenLambda :: Partition -> Map Partition Int #-} +{-# SPECIALIZE kostkaNumbersWithGivenLambda :: Partition -> Map Partition Integer #-} +kostkaNumbersWithGivenLambda :: forall coeff. Num coeff => Partition -> Map Partition coeff +kostkaNumbersWithGivenLambda plambda@(Partition lam) = evalState (worker lam) Map.empty where + + worker :: [Int] -> State (Map Partition (Map Partition coeff)) (Map Partition coeff) + worker unlam = case unlam of + [] -> return $ Map.singleton (Partition []) 1 + _ -> do + cache <- get + case Map.lookup (Partition unlam) cache of + Just sol -> return sol + Nothing -> do + let s = foldl' (+) 0 unlam + subsols <- forM (prevLambdas0 unlam) $ \p -> do + sub <- worker p + let t = s - foldl' (+) 0 p + f (Partition xs , c) = case xs of + (y:_) -> if t >= y then Just (Partition (t:xs) , c) else Nothing + [] -> if t > 0 then Just (Partition [t] , c) else Nothing + if t > 0 + then return $ Map.fromList $ mapMaybe f $ Map.toList sub + else return $ Map.empty + + let sol = Map.unionsWith (+) subsols + put $! (Map.insert (Partition unlam) sol cache) + return sol + + -- needs decreasing sequence + prevLambdas0 :: [Int] -> [[Int]] + prevLambdas0 (l:ls) = go l ls where + go b [a] = [ [x] | x <- [a..b] ] ++ [ [x,y] | x <- [a..b] , y<-[1..a] ] + go b (a:as) = [ x:xs | x <- [a..b] , xs <- go a as ] + go b [] = [] : [ [j] | j <- [1..b] ] + prevLambdas0 [] = [] + +-- | Lists all (positive) Kostka numbers @K(lambda,mu)@ with the given @mu@: +-- +-- > kostkaNumbersWithGivenMu mu == Map.fromList [ (lambda , kostkaNumber lambda mu) | lambda <- dominatingPartitions mu ] +-- +-- This function uses the iterated Pieri rule, and is relatively fast. +-- +kostkaNumbersWithGivenMu :: Partition -> Map Partition Int +kostkaNumbersWithGivenMu (Partition mu) = iteratedPieriRule (reverse mu) + +-------------------------------------------------------------------------------- +-- * Gelfand-Tsetlin patterns + +-- | A Gelfand-Tstetlin tableau +type GT = [[Int]] + +asciiGT :: GT -> ASCII +asciiGT gt = tabulate (HRight,VTop) (HSepSpaces 1, VSepEmpty) + $ (map . map) asciiShow + $ gt + +kostkaGelfandTsetlinPatterns :: Partition -> Partition -> [GT] +kostkaGelfandTsetlinPatterns lambda (Partition mu) = kostkaGelfandTsetlinPatterns' lambda mu + +-- | Generates all Kostka-Gelfand-Tsetlin tableau, that is, triangular arrays like +-- +-- > [ 3 ] +-- > [ 3 , 2 ] +-- > [ 3 , 1 , 0 ] +-- > [ 2 , 0 , 0 , 0 ] +-- +-- with both rows and column non-increasing such that +-- the top diagonal read lambda (in this case @lambda=[3,2]@) and the diagonal sums +-- are partial sums of mu (in this case @mu=[2,1,1,1]@) +-- +-- The number of such GT tableaux is the Kostka +-- number K(lambda,mu). +-- +kostkaGelfandTsetlinPatterns' :: Partition -> [Int] -> [GT] +kostkaGelfandTsetlinPatterns' plam@(Partition lambda0) mu0 + | minimum mu0 < 0 = [] + | wlam == 0 = if wmu == 0 then [ [] ] else [] + | wmu == wlam && plam `dominates` pmu = list + | otherwise = [] + where + + pmu = mkPartition mu0 + + nlam = length lambda0 + nmu = length mu0 + + n = max nlam nmu + + lambda = lambda0 ++ replicate (n - nlam) 0 + mu = mu0 ++ replicate (n - nmu ) 0 + + revlam = reverse lambda + + wmu = sum' mu + wlam = sum' lambda + + list = worker + revlam + (scanl1 (+) mu) + (replicate (n-1) 0) + (replicate (n ) 0) + [] + + worker + :: [Int] -- lambda_i in reverse order + -> [Int] -- partial sums of mu + -> [Int] -- sums of the tails of previous rows + -> [Int] -- last row + -> [[Int]] -- the lower part of GT tableau we accumulated so far (this is not needed if we only want to count) + -> [GT] + + worker (rl:rls) (smu:smus) (a:acc) (lastx0:lastrowt) table = stuff + where + x0 = smu - a + stuff = concat + [ worker rls smus (zipWith (+) acc (tail row)) (init row) (row:table) + | row <- boundedNonIncrSeqs' x0 (map (max rl) (max lastx0 x0 : lastrowt)) lambda + ] + worker [rl] _ _ _ table = [ [rl]:table ] + worker [] _ _ _ _ = [ [] ] + + boundedNonIncrSeqs' :: Int -> [Int] -> [Int] -> [[Int]] + boundedNonIncrSeqs' = go where + go h0 (a:as) (b:bs) = [ h:hs | h <- [(max 0 a)..(min h0 b)] , hs <- go h as bs ] + go _ [] _ = [[]] + go _ _ [] = [[]] + +-------------------------------------------------------------------------------- + +-- | This returns the corresponding Kostka number: +-- +-- > countKostkaGelfandTsetlinPatterns lambda mu == length (kostkaGelfandTsetlinPatterns lambda mu) +-- +countKostkaGelfandTsetlinPatterns :: Partition -> Partition -> Int +countKostkaGelfandTsetlinPatterns plam@(Partition lambda0) pmu@(Partition mu0) + | wlam == 0 = if wmu == 0 then 1 else 0 + | wmu == wlam && plam `dominates` pmu = cnt + | otherwise = 0 + where + + nlam = length lambda0 + nmu = length mu0 + + n = max nlam nmu + + lambda = lambda0 ++ replicate (n - nlam) 0 + mu = mu0 ++ replicate (n - nmu ) 0 + + revlam = reverse lambda + + wmu = sum' mu + wlam = sum' lambda + + cnt = worker + revlam + (scanl1 (+) mu) + (replicate (n-1) 0) + (replicate (n ) 0) + + worker + :: [Int] -- lambda_i in reverse order + -> [Int] -- partial sums of mu + -> [Int] -- sums of the tails of previous rows + -> [Int] -- last row + -> Int + + worker (rl:rls) (smu:smus) (a:acc) (lastx0:lastrowt) = stuff + where + x0 = smu - a + stuff = sum' + [ worker rls smus (zipWith (+) acc (tail row)) (init row) + | row <- boundedNonIncrSeqs' x0 (map (max rl) (max lastx0 x0 : lastrowt)) lambda + ] + worker [rl] _ _ _ = 1 + worker [] _ _ _ = 1 + + boundedNonIncrSeqs' :: Int -> [Int] -> [Int] -> [[Int]] + boundedNonIncrSeqs' = go where + go h0 (a:as) (b:bs) = [ h:hs | h <- [(max 0 a)..(min h0 b)] , hs <- go h as bs ] + go _ [] _ = [[]] + go _ _ [] = [[]] + +-------------------------------------------------------------------------------- + +{- + +-- | All non-increasing sentences between a lower and an upper bound +boundedNonIncrSeqs :: [Int] -> [Int] -> [[Int]] +boundedNonIncrSeqs as bs = case bs of + (h0:_) -> boundedNonIncrSeqs' h0 as bs + [] -> [[]] + +-- | All non-increasing sentences between a lower and an upper bound, and also less-or-equal than the given number +boundedNonIncrSeqs' :: Int -> [Int] -> [Int] -> [[Int]] +boundedNonIncrSeqs' = go where + go h0 (a:as) (b:bs) = [ h:hs | h <- [(max 0 a)..(min h0 b)] , hs <- go h as bs ] + go _ [] _ = [[]] + go _ _ [] = [[]] + +-- | All non-decreasing sentences between a lower and an upper bound +boundedNonDecrSeqs :: [Int] -> [Int] -> [[Int]] +boundedNonDecrSeqs = boundedNonDecrSeqs' 0 + +-- | All non-decreasing sentences between a lower and an upper bound, and also greator-or-equal then the given number +boundedNonDecrSeqs' :: Int -> [Int] -> [Int] -> [[Int]] +boundedNonDecrSeqs' h0 = go (max 0 h0) where + go h0 (a:as) (b:bs) = [ h:hs | h <- [(max h0 a)..b] , hs <- go h as bs ] + go _ [] _ = [[]] + go _ _ [] = [[]] + +-} + +-------------------------------------------------------------------------------- +-- * The iterated Pieri rule + +-- | Computes the Schur expansion of @h[n1]*h[n2]*h[n3]*...*h[nk]@ via iterating the Pieri rule. +-- Note: the coefficients are actually the Kostka numbers; the following is true: +-- +-- > Map.toList (iteratedPieriRule (fromPartition mu)) == [ (lam, kostkaNumber lam mu) | lam <- dominatingPartitions mu ] +-- +-- This should be faster than individually computing all these Kostka numbers. +-- +iteratedPieriRule :: Num coeff => [Int] -> Map Partition coeff +iteratedPieriRule = iteratedPieriRule' (Partition []) + +-- | Iterating the Pieri rule, we can compute the Schur expansion of +-- @h[lambda]*h[n1]*h[n2]*h[n3]*...*h[nk]@ +iteratedPieriRule' :: Num coeff => Partition -> [Int] -> Map Partition coeff +iteratedPieriRule' plambda ns = iteratedPieriRule'' (plambda,1) ns + +{-# SPECIALIZE iteratedPieriRule'' :: (Partition,Int ) -> [Int] -> Map Partition Int #-} +{-# SPECIALIZE iteratedPieriRule'' :: (Partition,Integer) -> [Int] -> Map Partition Integer #-} +iteratedPieriRule'' :: Num coeff => (Partition,coeff) -> [Int] -> Map Partition coeff +iteratedPieriRule'' (plambda,coeff0) ns = worker (Map.singleton plambda coeff0) ns where + worker old [] = old + worker old (n:ns) = worker new ns where + stuff = [ (coeff, pieriRule lam n) | (lam,coeff) <- Map.toList old ] + new = foldl' f Map.empty stuff + f t0 (c,ps) = foldl' (\t p -> Map.insertWith (+) p c t) t0 ps + +-------------------------------------------------------------------------------- + +-- | Computes the Schur expansion of @e[n1]*e[n2]*e[n3]*...*e[nk]@ via iterating the Pieri rule. +-- Note: the coefficients are actually the Kostka numbers; the following is true: +-- +-- > Map.toList (iteratedDualPieriRule (fromPartition mu)) == +-- > [ (dualPartition lam, kostkaNumber lam mu) | lam <- dominatingPartitions mu ] +-- +-- This should be faster than individually computing all these Kostka numbers. +-- It is a tiny bit slower than 'iteratedPieriRule'. +-- +iteratedDualPieriRule :: Num coeff => [Int] -> Map Partition coeff +iteratedDualPieriRule = iteratedDualPieriRule' (Partition []) + +-- | Iterating the Pieri rule, we can compute the Schur expansion of +-- @e[lambda]*e[n1]*e[n2]*e[n3]*...*e[nk]@ +iteratedDualPieriRule' :: Num coeff => Partition -> [Int] -> Map Partition coeff +iteratedDualPieriRule' plambda ns = iteratedDualPieriRule'' (plambda,1) ns + +{-# SPECIALIZE iteratedDualPieriRule'' :: (Partition,Int ) -> [Int] -> Map Partition Int #-} +{-# SPECIALIZE iteratedDualPieriRule'' :: (Partition,Integer) -> [Int] -> Map Partition Integer #-} +iteratedDualPieriRule'' :: Num coeff => (Partition,coeff) -> [Int] -> Map Partition coeff +iteratedDualPieriRule'' (plambda,coeff0) ns = worker (Map.singleton plambda coeff0) ns where + worker old [] = old + worker old (n:ns) = worker new ns where + stuff = [ (coeff, dualPieriRule lam n) | (lam,coeff) <- Map.toList old ] + new = foldl' f Map.empty stuff + f t0 (c,ps) = foldl' (\t p -> Map.insertWith (+) p c t) t0 ps + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Tableaux/GelfandTsetlin/Cone.hs b/Math/Combinat/Tableaux/GelfandTsetlin/Cone.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Tableaux/GelfandTsetlin/Cone.hs @@ -0,0 +1,261 @@ + +-- TODO: better name? + +-- | This module contains a function to generate (equivalence classes of) +-- triangular tableaux of size /k/, strictly increasing to the right and +-- to the bottom. For example +-- +-- > 1 +-- > 2 4 +-- > 3 5 8 +-- > 6 7 9 10 +-- +-- is such a tableau of size 4. +-- The numbers filling a tableau always consist of an interval @[1..c]@; +-- @c@ is called the /content/ of the tableaux. There is a unique tableau +-- of minimal content @2k-1@: +-- +-- > 1 +-- > 2 3 +-- > 3 4 5 +-- > 4 5 6 7 +-- +-- Let us call the tableaux with maximal content (that is, @m = binomial (k+1) 2@) +-- /standard/. The number of such standard tableaux are +-- +-- > 1, 1, 2, 12, 286, 33592, 23178480, ... +-- +-- OEIS:A003121, \"Strict sense ballot numbers\", +-- . +-- +-- See +-- R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88. +-- +-- The number of tableaux with content @c=m-d@ are +-- +-- > d= | 0 1 2 3 ... +-- > -----+---------------------------------------------- +-- > k=2 | 1 +-- > k=3 | 2 1 +-- > k=4 | 12 18 8 1 +-- > k=5 | 286 858 1001 572 165 22 1 +-- > k=6 | 33592 167960 361114 436696 326196 155584 47320 8892 962 52 1 +-- +-- We call these \"GT simplex tableaux\" (in the lack of a better name), since +-- they are in bijection with the simplicial cones in a canonical simplicial +-- decompositions of the Gelfand-Tsetlin cones (the content corresponds +-- to the dimension), which encode the combinatorics of Kostka numbers. +-- + +{-# LANGUAGE FlexibleInstances, TypeSynonymInstances #-} +module Math.Combinat.Tableaux.GelfandTsetlin.Cone + ( + -- * Types + Tableau + , Tri(..) + , TriangularArray + , fromTriangularArray + , triangularArrayUnsafe + -- * ASCII + , asciiTriangularArray + , asciiTableau + -- * Content + , gtSimplexContent + , _gtSimplexContent + , invertGTSimplexTableau + , _invertGTSimplexTableau + -- * Enumeration + , gtSimplexTableaux + , _gtSimplexTableaux + , countGTSimplexTableaux + ) + where + +-------------------------------------------------------------------------------- + +import Data.Ix +import Data.Ord +import Data.List + +import Control.Monad +import Control.Monad.ST +import Data.Array.IArray +import Data.Array.Unboxed +import Data.Array.ST + +import Math.Combinat.Tableaux (Tableau) +import Math.Combinat.Helper +import Math.Combinat.ASCII + +-------------------------------------------------------------------------------- + +-- | Triangular arrays +type TriangularArray a = Array Tri a + +-- | Set of @(i,j)@ pairs with @i>=j>=1@. +newtype Tri = Tri { unTri :: (Int,Int) } deriving (Eq,Ord,Show) + +binom2 :: Int -> Int +binom2 n = (n*(n-1)) `div` 2 + +index' :: Tri -> Int +index' (Tri (i,j)) = binom2 i + j - 1 + +-- it should be (1+8*m), +-- the 2 is a hack to be safe with the floating point stuff +deIndex' :: Int -> Tri +deIndex' m = Tri ( i+1 , m - binom2 (i+1) + 1 ) where + i = ( (floor.sqrt.(fromIntegral::Int->Double)) (2+8*m) - 1 ) `div` 2 + +instance Ix Tri where + index (a,b) x = index' x - index' a + inRange (a,b) x = (u<=j && j<=v) where + u = index' a + v = index' b + j = index' x + range (a,b) = map deIndex' [ index' a .. index' b ] + rangeSize (a,b) = index' b - index' a + 1 + +triangularArrayUnsafe :: Tableau a -> TriangularArray a +triangularArrayUnsafe tableau = listArray (Tri (1,1),Tri (k,k)) (concat tableau) + where k = length tableau + +fromTriangularArray :: TriangularArray a -> Tableau a +fromTriangularArray arr = (map.map) snd $ groupBy (equating f) $ assocs arr + where f = fst . unTri . fst + +-------------------------------------------------------------------------------- + +asciiTriangularArray :: Show a => TriangularArray a -> ASCII +asciiTriangularArray = asciiTableau . fromTriangularArray + +asciiTableau :: Show a => Tableau a -> ASCII +asciiTableau xxs = tabulate (HRight,VTop) (HSepSpaces 1, VSepEmpty) + $ (map . map) asciiShow + $ xxs + +instance Show a => DrawASCII (TriangularArray a) where + ascii = asciiTriangularArray + +-- instance Show a => DrawASCII (Tableau a) where +-- ascii = asciiTableau + +-------------------------------------------------------------------------------- + +-- "fractional fillings" +data Hole = Hole Int Int deriving (Eq,Ord,Show) + +type ReverseTableau = [[Int ]] +type ReverseHoleTableau = [[Hole]] + +toHole :: Int -> Hole +toHole k = Hole k 0 + +nextHole :: Hole -> Hole +nextHole (Hole k l) = Hole k (l+1) + +reverseTableau :: [[a]] -> [[a]] +reverseTableau = reverse . map reverse + +-------------------------------------------------------------------------------- + +gtSimplexContent :: TriangularArray Int -> Int +gtSimplexContent arr = max (arr ! (fst (bounds arr))) (arr ! (snd (bounds arr))) -- we also handle inverted tableau + +_gtSimplexContent :: Tableau Int -> Int +_gtSimplexContent t = max (head $ head t) (last $ last t) -- we also handle inverted tableau + +normalize :: ReverseHoleTableau -> TriangularArray Int +normalize = snd . normalize' + +-- returns ( content , tableau ) +normalize' :: ReverseHoleTableau -> ( Int , TriangularArray Int ) +normalize' holes = ( c , array (Tri (1,1), Tri (k,k)) xys ) where + k = length holes + c = length sorted + xys = concat $ zipWith hs [1..] sorted + hs a xs = map (h a) xs + h a (ij,_) = (Tri ij , a) + sorted = groupSortBy snd (concat withPos) + withPos = zipWith f [1..] (reverseTableau holes) + f i xs = zipWith (g i) [1..] xs + g i j hole = ((i,j),hole) + +-------------------------------------------------------------------------------- + +startHole :: [Hole] -> [Int] -> Hole +startHole (t:ts) (p:ps) = max t (toHole p) +startHole (t:ts) [] = t +startHole [] (p:ps) = toHole p +startHole [] [] = error "startHole" + +-- c is the "content" of the small tableau +enumHoles :: Int -> Hole -> [Hole] +enumHoles c start@(Hole k l) + = nextHole start + : [ Hole i 0 | i <- [k+1..c] ] ++ [ Hole i 1 | i <- [k+1..c] ] + +helper :: Int -> [Int] -> [Hole] -> [[Hole]] +helper c [] this = [[]] +helper c prev@(p:ps) this = + [ t:rest | t <- enumHoles c (startHole this prev), rest <- helper c ps (t:this) ] + +newLines' :: Int -> [Int] -> [[Hole]] +newLines' c lastReversed = helper c last [] + where + top = head lastReversed + last = reverse (top : lastReversed) + +newLines :: [Int] -> [[Hole]] +newLines lastReversed = newLines' (head lastReversed) lastReversed + +-- | Generates all tableaux of size @k@. Effective for @k<=6@. +gtSimplexTableaux :: Int -> [TriangularArray Int] +gtSimplexTableaux 0 = [ triangularArrayUnsafe [] ] +gtSimplexTableaux 1 = [ triangularArrayUnsafe [[1]] ] +gtSimplexTableaux k = map normalize $ concatMap f smalls where + smalls :: [ [[Int]] ] + smalls = map (reverseTableau . fromTriangularArray) $ gtSimplexTableaux (k-1) + f :: [[Int]] -> [ [[Hole]] ] + f small = map (:smallhole) $ map reverse $ newLines (head small) where + smallhole = map (map toHole) small + +_gtSimplexTableaux :: Int -> [Tableau Int] +_gtSimplexTableaux k = map fromTriangularArray $ gtSimplexTableaux k + +-------------------------------------------------------------------------------- + +-- | Note: This is slow (it actually generates all the tableaux) +countGTSimplexTableaux :: Int -> [Int] +countGTSimplexTableaux = elems . sizes' + +sizes' :: Int -> UArray Int Int +sizes' k = + runSTUArray $ do + let (a,b) = ( 2*k-1 , binom2 (k+1) ) + ar <- newArray (a,b) 0 :: ST s (STUArray s Int Int) + mapM_ (worker ar) $ gtSimplexTableaux k + return ar + where + worker :: STUArray s Int Int -> TriangularArray Int -> ST s () + worker ar t = do + let c = gtSimplexContent t + n <- readArray ar c + writeArray ar c (n+1) + +-------------------------------------------------------------------------------- + +-- | We can flip the numbers in the tableau so that the interval @[1..c]@ becomes +-- @[c..1]@. This way we a get a maybe more familiar form, when each row and each +-- column is strictly /decreasing/ (to the right and to the bottom). +invertGTSimplexTableau :: TriangularArray Int -> TriangularArray Int +invertGTSimplexTableau t = amap f t where + c = gtSimplexContent t + f x = c+1-x + +_invertGTSimplexTableau :: [[Int]] -> [[Int]] +_invertGTSimplexTableau t = (map . map) f t where + c = _gtSimplexContent t + f x = c+1-x + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Tableaux/LittlewoodRichardson.hs b/Math/Combinat/Tableaux/LittlewoodRichardson.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Tableaux/LittlewoodRichardson.hs @@ -0,0 +1,399 @@ + +-- | The Littlewood-Richardson rule + +module Math.Combinat.Tableaux.LittlewoodRichardson + ( lrCoeff , lrCoeff' + , lrMult + , lrRule , _lrRule , lrRuleNaive + , lrScalar , _lrScalar + ) + where + +-------------------------------------------------------------------------------- + +import Data.List +import Data.Maybe + +import Math.Combinat.Partitions.Integer +import Math.Combinat.Partitions.Skew +import Math.Combinat.Tableaux +import Math.Combinat.Tableaux.Skew +import Math.Combinat.Helper + +import Data.Map.Strict (Map) +import qualified Data.Map.Strict as Map + +-------------------------------------------------------------------------------- + +-- | Naive (very slow) reference implementation of the Littlewood-Richardson rule, based +-- on the definition "count the semistandard skew tableaux whose row content is a lattice word" +-- +lrRuleNaive :: SkewPartition -> Map Partition Int +lrRuleNaive skew = final where + n = skewPartitionWeight skew + ssst = semiStandardSkewTableaux n skew + final = foldl' f Map.empty $ catMaybes [ skewTableauRowContent skew | skew <- ssst ] + f old nu = Map.insertWith (+) nu 1 old + +-------------------------------------------------------------------------------- +-- SKEW EXPANSION + +-- | @lrRule@ computes the expansion of a skew Schur function +-- @s[lambda\/mu]@ via the Littlewood-Richardson rule. +-- +-- Adapted from John Stembridge's Maple code: +-- +-- +-- > lrRule (mkSkewPartition (lambda,nu)) == Map.fromList list where +-- > muw = weight lambda - weight nu +-- > list = [ (mu, coeff) +-- > | mu <- partitions muw +-- > , let coeff = lrCoeff lambda (mu,nu) +-- > , coeff /= 0 +-- > ] +-- +lrRule :: SkewPartition -> Map Partition Int +lrRule skew = _lrRule lam mu where + (lam,mu) = fromSkewPartition skew + +-- | @_lrRule lambda mu@ computes the expansion of the skew +-- Schur function @s[lambda\/mu]@ via the Littlewood-Richardson rule. +-- +_lrRule :: Partition -> Partition -> Map Partition Int +_lrRule plam@(Partition lam) pmu@(Partition mu0) = + if not (pmu `isSubPartitionOf` plam) + then Map.empty + else foldl' f Map.empty [ nu | (nu,_) <- fillings n diagram ] + where + f old nu = Map.insertWith (+) (Partition nu) 1 old + diagram = [ (i,j) | (i,a,b) <- reverse (zip3 [1..] lam mu) , j <- [b+1..a] ] + mu = mu0 ++ repeat 0 + n = sum' $ zipWith (-) lam mu -- n == length diagram + +{- +LR_rule:=proc(lambda) local l,mu,alpha,beta,i,j,dgrm; + if not `LR_rule/fit`(lambda,args[2]) then RETURN(0) fi; + l:=nops(lambda); mu:=[op(args[2]),0$l]; + dgrm:=[seq(seq([i,-j],j=-lambda[i]..-1-mu[i]),i=1..l)]; + if nargs>2 then alpha:=args[3]; + if nargs>3 then beta:=args[4] else beta:=[] fi; + if not `LR_rule/fit`(alpha,beta) then RETURN(0) fi; + l:=convert([op(lambda),op(beta)],`+`); + if l<>convert([op(alpha),op(mu)],`+`) then RETURN(0) fi; + nops(LR_fillings(dgrm,[alpha,beta])) + else + convert([seq(s[op(i[1])],i=LR_fillings(dgrm))],`+`) + fi +end; +-} + +-------------------------------------------------------------------------------- + +-- | A filling is a pair consisting a shape @nu@ and a lattice permutation @lp@ +type Filling = ( [Int] , [Int] ) + +-- | A diagram is a set of boxes in a skew shape (in the right order) +type Diagram = [ (Int,Int) ] + +-- | Note: we use reverse ordering in Diagrams compared the Stembridge's code. +-- Also, for performance reasons, we need the length of the diagram +fillings :: Int -> Diagram -> [Filling] +fillings _ [] = [ ([],[]) ] +fillings n diagram@((x,y):rest) = concatMap (nextLetter lower upper) (fillings (n-1) rest) where + upper = case findIndex (==(x ,y+1)) diagram of { Just j -> n-j ; Nothing -> 0 } + lower = case findIndex (==(x-1,y )) diagram of { Just j -> n-j ; Nothing -> 0 } + +{- +LR_fillings:=proc(dgrm) local n,x,upper,lower; + if dgrm=[] then + if nargs=1 then x:=[] else x:=args[2][2] fi; + RETURN([[x,[]]]) + fi; + n:=nops(dgrm); x:=dgrm[n]; + if not member([x[1],x[2]+1],dgrm,'upper') then upper:=0 fi; + if not member([x[1]-1,x[2]],dgrm,'lower') then lower:=0 fi; + if nargs=1 then + map(`LR/nextletter`,LR_fillings([op(1..n-1,dgrm)]),lower,upper) + else + map(`LR/nextletter`,LR_fillings([op(1..n-1,dgrm)],args[2]), + lower,upper,args[2][1]) + fi; +end: +-} + +-------------------------------------------------------------------------------- + +nextLetter :: Int -> Int -> Filling -> [Filling] +nextLetter lower upper (nu,lpart) = stuff where + stuff = [ ( incr i shape , lpart++[i] ) | i<-nlist ] + shape = nu ++ [0] + lb = if lower>0 + then lpart !! (lower-1) + else 0 + ub = if upper>0 + then min (length shape) (lpart !! (upper-1)) + else length shape + + nlist = filter (>0) $ map f [lb+1..ub] + f j = if j==1 || shape!!(j-2) > shape!!(j-1) then j else 0 + +{- + -- another nlist implementation, but doesn't seem to be faster + (h0:hs0) = drop lb (-666:shape) + nlist = go h0 hs0 [lb+1..ub] where + go !lasth (h:hs) (j:js) = if j==1 || lasth > h + then j : go h hs js + else go h hs js + go _ _ [] = [] +-} + + -- increments the i-th element (starting from 1) + incr :: Int -> [Int] -> [Int] + incr i (x:xs) = case i of + 0 -> finish (x:xs) + 1 -> (x+1) : finish xs + _ -> x : incr (i-1) xs + incr _ [] = [] + + -- removes tailing zeros + finish :: [Int] -> [Int] + finish (x:xs) = if x>0 then x : finish xs else [] + finish [] = [] + +{- +`LR/nextletter`:=proc(T) local shape,lp,lb,ub,i,nl; + shape:=[op(T[1]),0]; lp:=T[2]; ub:=nops(shape); + if nargs>3 then ub:=min(ub,nops(args[4])) fi; + if args[2]=0 then lb:=0 else lb:=lp[args[2]] fi; + if args[3]>0 then ub:=min(lp[args[3]],ub) fi; + if nargs<4 then + nl:=map(proc(x,y) if x=1 or y[x-1]>y[x] then x fi end,[$lb+1..ub],shape) + else + nl:=map(proc(x,y,z) if y[x]y[x]) then x fi end, + [$lb+1..ub],shape,args[4]) + fi; + nl:=[seq([subsop(i=shape[i]+1,shape),[op(lp),i]],i=nl)]; + op(subs(0=NULL,nl)) +end: +-} + +-------------------------------------------------------------------------------- +-- COEFF + +-- | @lrCoeff lam (mu,nu)@ computes the coressponding Littlewood-Richardson coefficients. +-- This is also the coefficient of @s[lambda]@ in the product @s[mu]*s[nu]@ +-- +-- Note: This is much slower than using 'lrRule' or 'lrMult' to compute several coefficients +-- at the same time! +lrCoeff :: Partition -> (Partition,Partition) -> Int +lrCoeff lam (mu,nu) = + if nu `isSubPartitionOf` lam + then lrScalar (mkSkewPartition (lam,nu)) (mkSkewPartition (mu,emptyPartition)) + else 0 + +-- | @lrCoeff (lam\/nu) mu@ computes the coressponding Littlewood-Richardson coefficients. +-- This is also the coefficient of @s[mu]@ in the product @s[lam\/nu]@ +-- +-- Note: This is much slower than using 'lrRule' or 'lrMult' to compute several coefficients +-- at the same time! +lrCoeff' :: SkewPartition -> Partition -> Int +lrCoeff' skew p = lrScalar skew (mkSkewPartition (p,emptyPartition)) + +-------------------------------------------------------------------------------- +-- SCALAR PRODUCT + +-- | @lrScalar (lambda\/mu) (alpha\/beta)@ computes the scalar product of the two skew +-- Schur functions @s[lambda\/mu]@ and @s[alpha\/beta]@ via the Littlewood-Richardson rule. +-- +-- Adapted from John Stembridge Maple code: +-- +-- +lrScalar :: SkewPartition -> SkewPartition -> Int +lrScalar lambdaMu alphaBeta = _lrScalar (fromSkewPartition lambdaMu) (fromSkewPartition alphaBeta) + +_lrScalar :: (Partition,Partition) -> (Partition,Partition) -> Int +_lrScalar (plam @(Partition lam ) , pmu @(Partition mu0) ) + (palpha@(Partition alpha) , pbeta@(Partition beta)) = + if not (pmu `isSubPartitionOf` plam ) + || not (pbeta `isSubPartitionOf` palpha) + || (sum' lam + sum' beta) /= (sum' alpha + sum' mu0) -- equivalent to (lambda-mu) /= (alpha-beta) + then 0 + else length $ fillings' n diagram (alpha,beta) + where + f old nu = Map.insertWith (+) (Partition nu) 1 old + diagram = [ (i,j) | (i,a,b) <- reverse (zip3 [1..] lam mu) , j <- [b+1..a] ] + mu = mu0 ++ repeat 0 + n = sum' $ zipWith (-) lam mu -- n == length diagram + +{- +LR_rule:=proc(lambda) local l,mu,alpha,beta,i,j,dgrm; + if not `LR_rule/fit`(lambda,args[2]) then RETURN(0) fi; + l:=nops(lambda); mu:=[op(args[2]),0$l]; + dgrm:=[seq(seq([i,-j],j=-lambda[i]..-1-mu[i]),i=1..l)]; + if nargs>2 then alpha:=args[3]; + if nargs>3 then beta:=args[4] else beta:=[] fi; + if not `LR_rule/fit`(alpha,beta) then RETURN(0) fi; + l:=convert([op(lambda),op(beta)],`+`); + if l<>convert([op(alpha),op(mu)],`+`) then RETURN(0) fi; + nops(LR_fillings(dgrm,[alpha,beta])) + else + convert([seq(s[op(i[1])],i=LR_fillings(dgrm))],`+`) + fi +end; +-} + +-------------------------------------------------------------------------------- + +-- | Note: we use reverse ordering in Diagrams compared the Stembridge's code. +-- Also, for performance reasons, we need the length of the diagram +fillings' :: Int -> Diagram -> ([Int],[Int]) -> [Filling] +fillings' _ [] (alpha,beta) = [ (beta,[]) ] +fillings' n diagram@((x,y):rest) alphaBeta@(alpha,beta) = stuff where + stuff = concatMap (nextLetter' lower upper alpha) (fillings' (n-1) rest alphaBeta) + upper = case findIndex (==(x ,y+1)) diagram of { Just j -> n-j ; Nothing -> 0 } + lower = case findIndex (==(x-1,y )) diagram of { Just j -> n-j ; Nothing -> 0 } + +{- +LR_fillings:=proc(dgrm) local n,x,upper,lower; + if dgrm=[] then + if nargs=1 then x:=[] else x:=args[2][2] fi; + RETURN([[x,[]]]) + fi; + n:=nops(dgrm); x:=dgrm[n]; + if not member([x[1],x[2]+1],dgrm,'upper') then upper:=0 fi; + if not member([x[1]-1,x[2]],dgrm,'lower') then lower:=0 fi; + if nargs=1 then + map(`LR/nextletter`,LR_fillings([op(1..n-1,dgrm)]),lower,upper) + else + map(`LR/nextletter`,LR_fillings([op(1..n-1,dgrm)],args[2]), + lower,upper,args[2][1]) + fi; +end: +-} + +-------------------------------------------------------------------------------- + +nextLetter' :: Int -> Int -> [Int] -> Filling -> [Filling] +nextLetter' lower upper alpha (nu,lpart) = stuff where + stuff = [ ( incr i shape , lpart++[i] ) | i<-nlist ] + shape = nu ++ [0] + lb = if lower>0 + then lpart !! (lower-1) + else 0 + ub1 = if upper>0 + then min (length shape) (lpart !! (upper-1)) + else length shape + ub = min (length alpha) ub1 + nlist = filter (>0) $ map f [lb+1..ub] + f j = if ( shape!!(j-1) < alpha!!(j-1)) && + (j==1 || shape!!(j-2) > shape!!(j-1)) + then j + else 0 + + -- increments the i-th element (starting from 1) + incr :: Int -> [Int] -> [Int] + incr i (x:xs) = case i of + 0 -> finish (x:xs) + 1 -> (x+1) : finish xs + _ -> x : incr (i-1) xs + incr _ [] = [] + + -- removes tailing zeros + finish :: [Int] -> [Int] + finish (x:xs) = if x>0 then x : finish xs else [] + finish [] = [] + +{- +`LR/nextletter`:=proc(T) local shape,lp,lb,ub,i,nl; + shape:=[op(T[1]),0]; lp:=T[2]; ub:=nops(shape); + if nargs>3 then ub:=min(ub,nops(args[4])) fi; + if args[2]=0 then lb:=0 else lb:=lp[args[2]] fi; + if args[3]>0 then ub:=min(lp[args[3]],ub) fi; + if nargs<4 then + nl:=map(proc(x,y) if x=1 or y[x-1]>y[x] then x fi end,[$lb+1..ub],shape) + else + nl:=map(proc(x,y,z) if y[x]y[x]) then x fi end, + [$lb+1..ub],shape,args[4]) + fi; + nl:=[seq([subsop(i=shape[i]+1,shape),[op(lp),i]],i=nl)]; + op(subs(0=NULL,nl)) +end: +-} + +-------------------------------------------------------------------------------- +-- MULTIPLICATION + +type Part = [Int] + +-- | Computes the expansion of the product of Schur polynomials @s[mu]*s[nu]@ using the +-- Littlewood-Richardson rule. Note: this is symmetric in the two arguments. +-- +-- Based on the wikipedia article +-- +-- > lrMult mu nu == Map.fromList list where +-- > lamw = weight nu + weight mu +-- > list = [ (lambda, coeff) +-- > | lambda <- partitions lamw +-- > , let coeff = lrCoeff lambda (mu,nu) +-- > , coeff /= 0 +-- > ] +-- +lrMult :: Partition -> Partition -> Map Partition Int +lrMult pmu@(Partition mu) pnu@(Partition nu) = result where + result = foldl' add Map.empty (addMu mu nu) where + add !old lambda = Map.insertWith (+) (Partition lambda) 1 old + +-- | This basically lists all the outer shapes (with multiplicities) which can be result from the LR rule +addMu :: Part -> Part -> [Part] +addMu mu part = go ubs0 mu dmu part where + + go _ [] _ part = [part] + go ubs (m:ms) (d:ds) part = concat [ go (drop d ubs') ms ds part' | (ubs',part') <- addRowOf ubs part ] + + ubs0 = take (headOrZero mu) [headOrZero part + 1..] + dmu = diffSeq mu + + +-- | Adds a full row of @(length pcols)@ boxes containing to a partition, with +-- pcols being the upper bounds of the columns, respectively. We also return the +-- newly added columns +addRowOf :: [Int] -> Part -> [([Int],Part)] +addRowOf pcols part = go 0 pcols part [] where + go !lb [] p ncols = [(reverse ncols , p)] + go !lb (!ub:ubs) p ncols = concat [ go col ubs (addBox ij p) (col:ncols) | ij@(row,col) <- newBoxes (lb+1) ub p ] + +-- | Returns the (row,column) pairs of the new boxes which +-- can be added to the given partition with the given column bounds +-- and the 1-Rieri rule +newBoxes :: Int -> Int -> Part -> [(Int,Int)] +newBoxes lb ub part = reverse $ go [1..] part (headOrZero part + 1) where + go (!i:_ ) [] !lp + | lb <= 1 && 1 <= ub && lp > 0 = [(i,1)] + | otherwise = [] + go (!i:is) (!j:js) !lp + | j1 < lb = [] + | j1 <= ub && lp > j = (i,j1) : go is js j + | otherwise = go is js j + where + j1 = j+1 + +-- | Adds a box to a partition +addBox :: (Int,Int) -> Part -> Part +addBox (k,_) part = go 1 part where + go !i (p:ps) = if i==k then (p+1):ps else p : go (i+1) ps + go !i [] = if i==k then [1] else error "addBox: shouldn't happen" + +-- | Safe head defaulting to zero +headOrZero :: [Int] -> Int +headOrZero xs = case xs of + (!x:_) -> x + [] -> 0 + +-- | Computes the sequence of differences from a partition (including the last difference to zero) +diffSeq :: Part -> [Int] +diffSeq = go where + go (p:ps@(q:_)) = (p-q) : go ps + go [p] = [p] + go [] = [] + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Tableaux/Skew.hs b/Math/Combinat/Tableaux/Skew.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Tableaux/Skew.hs @@ -0,0 +1,223 @@ + +-- | Skew tableaux are skew partitions filled with numbers. +-- +-- For example: +-- +-- <> +-- + +{-# LANGUAGE CPP, BangPatterns, ScopedTypeVariables, MultiParamTypeClasses #-} + +module Math.Combinat.Tableaux.Skew where + +-------------------------------------------------------------------------------- + +import Data.List + +import Math.Combinat.Classes +import Math.Combinat.Partitions.Integer +import Math.Combinat.Partitions.Skew +import Math.Combinat.Tableaux +import Math.Combinat.ASCII +import Math.Combinat.Helper + +import Data.Map.Strict (Map) +import qualified Data.Map.Strict as Map + +-------------------------------------------------------------------------------- +-- * Basics +-- | A skew tableau is represented by a list of offsets and entries +newtype SkewTableau a = SkewTableau [(Int,[a])] deriving (Eq,Ord,Show) + +-- unSkewTableau :: SkewTableau a -> [(Int,[a])] +-- unSkewTableau (SkewTableau a) = a + +instance Functor SkewTableau where + fmap f (SkewTableau t) = SkewTableau [ (a, map f xs) | (a,xs) <- t ] + +-- | The shape of a skew tableau +skewTableauShape :: SkewTableau a -> SkewPartition +skewTableauShape (SkewTableau list) = SkewPartition [ (o,length xs) | (o,xs) <- list ] + +instance HasShape (SkewTableau a) SkewPartition where + shape = skewTableauShape + +-- | The weight of a tableau is the weight of its shape, or the number of entries +skewTableauWeight :: SkewTableau a -> Int +skewTableauWeight = skewPartitionWeight . skewTableauShape + +instance HasWeight (SkewTableau a) where + weight = skewTableauWeight + +-------------------------------------------------------------------------------- + +-- | The dual of a skew tableau, that is, its mirror image to the main diagonal +dualSkewTableau :: forall a. SkewTableau a -> SkewTableau a +dualSkewTableau (SkewTableau axs) = SkewTableau (go axs) where + + go [] = [] + go axs = case sub 0 axs of + (0,[]) -> [] + this -> this : go (strip axs) + + strip :: [(Int,[a])] -> [(Int,[a])] + strip [] = [] + strip ((a,xs):rest) = if a>0 + then (a-1,xs) : strip rest + else case xs of + [] -> [] + (z:zs) -> case zs of + [] -> [] + _ -> (0,zs) : strip rest + + sub :: Int -> [(Int,[a])] -> (Int,[a]) + sub !b [] = (b,[]) + sub !b ((a,this):rest) = if a>0 + then sub (b+1) rest + else (b,ys) where + ys = map head $ takeWhile (not . null) (this : map snd rest) + +{- +test_dualSkewTableau :: [SkewTableau Int] +test_dualSkewTableau = bad where + ps = allPartitions 11 + bad = [ st + | p<-ps , q<-ps + , (q `isSubPartitionOf` p) + , let sp = mkSkewPartition (p,q) + , let st = fillSkewPartitionWithRowWord sp [1..] + , dualSkewTableau (dualSkewTableau st) /= st + ] +-} + +instance HasDuality (SkewTableau a) where + dual = dualSkewTableau + +-------------------------------------------------------------------------------- +-- * Semistandard tableau + +-- | A tableau is /semistandard/ if its entries are weekly increasing horizontally +-- and strictly increasing vertically +isSemiStandardSkewTableau :: SkewTableau Int -> Bool +isSemiStandardSkewTableau st@(SkewTableau axs) = weak && strict where + weak = and [ isWeaklyIncreasing xs | (a,xs) <- axs ] + strict = and [ isStrictlyIncreasing ys | (b,ys) <- bys ] + SkewTableau bys = dualSkewTableau st + +-- | A tableau is /standard/ if it is semistandard and its content is exactly @[1..n]@, +-- where @n@ is the weight. +isStandardSkewTableau :: SkewTableau Int -> Bool +isStandardSkewTableau st = isSemiStandardSkewTableau st && sort (skewTableauRowWord st) == [1..n] where + n = skewTableauWeight st + +-------------------------------------------------------------------------------- + +-- | All semi-standard skew tableaux filled with the numbers @[1..n]@ +semiStandardSkewTableaux :: Int -> SkewPartition -> [SkewTableau Int] +semiStandardSkewTableaux n (SkewPartition abs) = map SkewTableau stuff where + + stuff = worker as bs ds (repeat 1) + (as,bs) = unzip abs + ds = diffSequence as + + -- | @worker inner outerMinusInner innerdiffs lowerbound + worker :: [Int] -> [Int] -> [Int] -> [Int] -> [[(Int,[Int])]] + worker (a:as) (b:bs) (d:ds) lb = [ (a,this):rest + | this <- row b 1 lb + , let lb' = (replicate d 1 ++ map (+1) this) + , rest <- worker as bs ds lb' ] + worker [] _ _ _ = [ [] ] + + -- @row length minimum lowerbound@ + row 0 _ _ = [[]] + row _ _ [] = [] + row !k !m (!a:as) = [ x:xs | x <- [(max a m)..n] , xs <- row (k-1) x as ] + +{- +-- | from a sequence @[a1,a2,..,an]@ computes the sequence of differences +-- @[a1-a2,a2-a3,...,an-0]@ +diffSequence :: [Int] -> [Int] +diffSequence = go where + go (x:ys@(y:_)) = (x-y) : go ys + go [x] = [x] + go [] = [] +-} + +-------------------------------------------------------------------------------- +-- * ASCII + +-- | ASCII drawing of a skew tableau (using the English notation) +asciiSkewTableau :: Show a => SkewTableau a -> ASCII +asciiSkewTableau = asciiSkewTableau' "." EnglishNotation + +asciiSkewTableau' + :: Show a + => String -- ^ string representing the elements of the inner (unfilled) partition + -> PartitionConvention -- ^ orientation + -> SkewTableau a + -> ASCII +asciiSkewTableau' innerstr orient (SkewTableau axs) = tabulate (HRight,VTop) (HSepSpaces 1, VSepEmpty) stuff where + stuff = case orient of + EnglishNotation -> es + EnglishNotationCCW -> reverse (transpose es) + FrenchNotation -> reverse es + inner = asciiFromString innerstr + es = [ replicate a inner ++ map asciiShow xs | (a,xs) <- axs ] + +instance Show a => DrawASCII (SkewTableau a) where + ascii = asciiSkewTableau + +-------------------------------------------------------------------------------- +-- * Row \/ column words + +-- | The reversed (right-to-left) rows, concatenated +skewTableauRowWord :: SkewTableau a -> [a] +skewTableauRowWord (SkewTableau axs) = concatMap (reverse . snd) axs + +-- | The reversed (bottom-to-top) columns, concatenated +skewTableauColumnWord :: SkewTableau a -> [a] +skewTableauColumnWord = skewTableauRowWord . dualSkewTableau + +-- | Fills a skew partition with content, in row word order +fillSkewPartitionWithRowWord :: SkewPartition -> [a] -> SkewTableau a +fillSkewPartitionWithRowWord (SkewPartition abs) xs = SkewTableau $ go abs xs where + go ((b,a):rest) xs = let (ys,zs) = splitAt a xs in (b,reverse ys) : go rest zs + go [] xs = [] + +-- | Fills a skew partition with content, in column word order +fillSkewPartitionWithColumnWord :: SkewPartition -> [a] -> SkewTableau a +fillSkewPartitionWithColumnWord shape content + = dualSkewTableau + $ fillSkewPartitionWithRowWord (dualSkewPartition shape) content + +-------------------------------------------------------------------------------- + +-- | If the skew tableau's row word is a lattice word, we can make a partition from its content +skewTableauRowContent :: SkewTableau Int -> Maybe Partition +skewTableauRowContent (SkewTableau axs) = go Map.empty rowword where + + rowword = concatMap (reverse . snd) axs + + finish table = Partition (f 1) where + f !i = case lkp i of + 0 -> [] + y -> y : f (i+1) + lkp j = case Map.lookup j table of + Just k -> k + Nothing -> 0 + + go :: Map Int Int -> [Int] -> Maybe Partition + go !table [] = Just (finish table) + go !table (i:is) = + if check i + then go table' is + else Nothing + where + table' = Map.insertWith (+) i 1 table + check j = j==1 || cnt (j-1) >= cnt j + cnt j = case Map.lookup j table' of + Just k -> k + Nothing -> 0 + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Trees.hs b/Math/Combinat/Trees.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Trees.hs @@ -0,0 +1,9 @@ + +module Math.Combinat.Trees + ( module Math.Combinat.Trees.Binary + , module Math.Combinat.Trees.Nary + ) where + +import Math.Combinat.Trees.Binary +import Math.Combinat.Trees.Nary + diff --git a/Math/Combinat/Trees/Binary.hs b/Math/Combinat/Trees/Binary.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Trees/Binary.hs @@ -0,0 +1,491 @@ + +-- | Binary trees, forests, etc. See: +-- Donald E. Knuth: The Art of Computer Programming, vol 4, pre-fascicle 4A. +-- +-- For example, here are all the binary trees on 4 nodes: +-- +-- <> +-- + +{-# LANGUAGE FlexibleInstances, TypeSynonymInstances #-} +module Math.Combinat.Trees.Binary + ( -- * Types + BinTree(..) + , leaf + , graft + , BinTree'(..) + , forgetNodeDecorations + , Paren(..) + , parenthesesToString + , stringToParentheses + , numberOfNodes + , numberOfLeaves + -- * Conversion to rose trees (@Data.Tree@) + , toRoseTree , toRoseTree' + , module Data.Tree + -- * Enumerate leaves + , enumerateLeaves_ + , enumerateLeaves + , enumerateLeaves' + -- * Nested parentheses + , nestedParentheses + , randomNestedParentheses + , nthNestedParentheses + , countNestedParentheses + , fasc4A_algorithm_P + , fasc4A_algorithm_W + , fasc4A_algorithm_U + -- * Generating binary trees + , binaryTrees + , countBinaryTrees + , binaryTreesNaive + , randomBinaryTree + , fasc4A_algorithm_R + -- * ASCII drawing + , asciiBinaryTree_ + -- * Graphviz drawing + , Dot + , graphvizDotBinTree + , graphvizDotBinTree' + , graphvizDotForest + , graphvizDotTree + -- * Bijections + , forestToNestedParentheses + , forestToBinaryTree + , nestedParenthesesToForest + , nestedParenthesesToForestUnsafe + , nestedParenthesesToBinaryTree + , nestedParenthesesToBinaryTreeUnsafe + , binaryTreeToForest + , binaryTreeToNestedParentheses + ) + where + +-------------------------------------------------------------------------------- + +import Control.Applicative +import Control.Monad +import Control.Monad.ST + +import Data.Array +import Data.Array.ST +import Data.Array.Unsafe + +import Data.List +import Data.Tree (Tree(..),Forest(..)) + +import Data.Monoid +import Data.Foldable (Foldable(foldMap)) +import Data.Traversable (Traversable(traverse)) + +import System.Random + +import Math.Combinat.Numbers (factorial,binomial) + +import Math.Combinat.Trees.Graphviz + ( Dot + , graphvizDotBinTree , graphvizDotBinTree' + , graphvizDotForest , graphvizDotTree + ) +import Math.Combinat.Classes +import Math.Combinat.Helper +import Math.Combinat.ASCII as ASCII + +-------------------------------------------------------------------------------- +-- * Types + +-- | A binary tree with leaves decorated with type @a@. +data BinTree a + = Branch (BinTree a) (BinTree a) + | Leaf a + deriving (Eq,Ord,Show,Read) + +leaf :: BinTree () +leaf = Leaf () + +-- | The monadic join operation of binary trees +graft :: BinTree (BinTree a) -> BinTree a +graft = go where + go (Branch l r) = Branch (go l) (go r) + go (Leaf t ) = t + +-------------------------------------------------------------------------------- + +-- | A binary tree with leaves and internal nodes decorated +-- with types @a@ and @b@, respectively. +data BinTree' a b + = Branch' (BinTree' a b) b (BinTree' a b) + | Leaf' a + deriving (Eq,Ord,Show,Read) + +forgetNodeDecorations :: BinTree' a b -> BinTree a +forgetNodeDecorations = go where + go (Branch' left _ right) = Branch (go left) (go right) + go (Leaf' decor ) = Leaf decor + +-------------------------------------------------------------------------------- + +instance HasNumberOfNodes (BinTree a) where + numberOfNodes = go where + go (Leaf _ ) = 0 + go (Branch l r) = go l + go r + 1 + +instance HasNumberOfLeaves (BinTree a) where + numberOfLeaves = go where + go (Leaf _ ) = 1 + go (Branch l r) = go l + go r + + +instance HasNumberOfNodes (BinTree' a b) where + numberOfNodes = go where + go (Leaf' _ ) = 0 + go (Branch' l _ r) = go l + go r + 1 + +instance HasNumberOfLeaves (BinTree' a b) where + numberOfLeaves = go where + go (Leaf' _ ) = 1 + go (Branch' l _ r) = go l + go r + +-------------------------------------------------------------------------------- +-- * Enumerate leaves + +-- | Enumerates the leaves a tree, starting from 0, ignoring old labels +enumerateLeaves_ :: BinTree a -> BinTree Int +enumerateLeaves_ = snd . go 0 where + go !k t = case t of + Leaf _ -> (k+1 , Leaf k) + Branch l r -> (k'', Branch l' r') where + (k' ,l') = go k l + (k'',r') = go k' r + +-- | Enumerates the leaves a tree, starting from zero, and also returns the number of leaves +enumerateLeaves' :: BinTree a -> (Int, BinTree (a,Int)) +enumerateLeaves' = go 0 where + go !k t = case t of + Leaf y -> (k+1 , Leaf (y,k)) + Branch l r -> (k'', Branch l' r') where + (k' ,l') = go k l + (k'',r') = go k' r + +-- | Enumerates the leaves a tree, starting from zero +enumerateLeaves :: BinTree a -> BinTree (a,Int) +enumerateLeaves = snd . enumerateLeaves' + +-------------------------------------------------------------------------------- +-- * conversion to 'Data.Tree' + +-- | Convert a binary tree to a rose tree (from "Data.Tree") +toRoseTree :: BinTree a -> Tree (Maybe a) +toRoseTree = go where + go (Branch t1 t2) = Node Nothing [go t1, go t2] + go (Leaf x) = Node (Just x) [] + +toRoseTree' :: BinTree' a b -> Tree (Either b a) +toRoseTree' = go where + go (Branch' t1 y t2) = Node (Left y) [go t1, go t2] + go (Leaf' x) = Node (Right x) [] + +-------------------------------------------------------------------------------- +-- instances + +instance Functor BinTree where + fmap f = go where + go (Branch left right) = Branch (go left) (go right) + go (Leaf x) = Leaf (f x) + +instance Foldable BinTree where + foldMap f = go where + go (Leaf x) = f x + go (Branch left right) = (go left) `mappend` (go right) + +instance Traversable BinTree where + traverse f = go where + go (Leaf x) = Leaf <$> f x + go (Branch left right) = Branch <$> go left <*> go right + +instance Applicative BinTree where + pure = Leaf + u <*> t = go u where + go (Branch l r) = Branch (go l) (go r) + go (Leaf f ) = fmap f t + +instance Monad BinTree where + return = Leaf + (>>=) t f = go t where + go (Branch l r) = Branch (go l) (go r) + go (Leaf y ) = f y + +-------------------------------------------------------------------------------- +-- * Nested parentheses + +data Paren + = LeftParen + | RightParen + deriving (Eq,Ord,Show,Read) + +parenToChar :: Paren -> Char +parenToChar LeftParen = '(' +parenToChar RightParen = ')' + +parenthesesToString :: [Paren] -> String +parenthesesToString = map parenToChar + +stringToParentheses :: String -> [Paren] +stringToParentheses [] = [] +stringToParentheses (x:xs) = p : stringToParentheses xs where + p = case x of + '(' -> LeftParen + ')' -> RightParen + _ -> error "stringToParentheses: invalid character" + +-------------------------------------------------------------------------------- +-- * Bijections + +forestToNestedParentheses :: Forest a -> [Paren] +forestToNestedParentheses = forest where + -- forest :: Forest a -> [Paren] + forest = concatMap tree + -- tree :: Tree a -> [Paren] + tree (Node _ sf) = LeftParen : forest sf ++ [RightParen] + +forestToBinaryTree :: Forest a -> BinTree () +forestToBinaryTree = forest where + -- forest :: Forest a -> BinTree () + forest = foldr Branch leaf . map tree + -- tree :: Tree a -> BinTree () + tree (Node _ sf) = case sf of + [] -> leaf + _ -> forest sf + +nestedParenthesesToForest :: [Paren] -> Maybe (Forest ()) +nestedParenthesesToForest ps = + case parseForest ps of + (rest,forest) -> case rest of + [] -> Just forest + _ -> Nothing + where + parseForest :: [Paren] -> ( [Paren] , Forest () ) + parseForest ps = unfoldEither parseTree ps + parseTree :: [Paren] -> Either [Paren] ( [Paren] , Tree () ) + parseTree orig@(LeftParen:ps) = let (rest,ts) = parseForest ps in case rest of + (RightParen:qs) -> Right (qs, Node () ts) + _ -> Left orig + parseTree qs = Left qs + +nestedParenthesesToForestUnsafe :: [Paren] -> Forest () +nestedParenthesesToForestUnsafe = fromJust . nestedParenthesesToForest + +nestedParenthesesToBinaryTree :: [Paren] -> Maybe (BinTree ()) +nestedParenthesesToBinaryTree ps = + case parseForest ps of + (rest,forest) -> case rest of + [] -> Just forest + _ -> Nothing + where + parseForest :: [Paren] -> ( [Paren] , BinTree () ) + parseForest ps = let (rest,ts) = unfoldEither parseTree ps in (rest , foldr Branch leaf ts) + parseTree :: [Paren] -> Either [Paren] ( [Paren] , BinTree () ) + parseTree orig@(LeftParen:ps) = let (rest,ts) = parseForest ps in case rest of + (RightParen:qs) -> Right (qs, ts) + _ -> Left orig + parseTree qs = Left qs + +nestedParenthesesToBinaryTreeUnsafe :: [Paren] -> BinTree () +nestedParenthesesToBinaryTreeUnsafe = fromJust . nestedParenthesesToBinaryTree + +binaryTreeToNestedParentheses :: BinTree a -> [Paren] +binaryTreeToNestedParentheses = worker where + worker (Branch l r) = LeftParen : worker l ++ RightParen : worker r + worker (Leaf _) = [] + +binaryTreeToForest :: BinTree a -> Forest () +binaryTreeToForest = worker where + worker (Branch l r) = Node () (worker l) : worker r + worker (Leaf _) = [] + +-------------------------------------------------------------------------------- +-- * Nested parentheses + +-- | Generates all sequences of nested parentheses of length @2n@ in +-- lexigraphic order. +-- +-- Synonym for 'fasc4A_algorithm_P'. +-- +nestedParentheses :: Int -> [[Paren]] +nestedParentheses = fasc4A_algorithm_P + +-- | Synonym for 'fasc4A_algorithm_W'. +randomNestedParentheses :: RandomGen g => Int -> g -> ([Paren],g) +randomNestedParentheses = fasc4A_algorithm_W + +-- | Synonym for 'fasc4A_algorithm_U'. +nthNestedParentheses :: Int -> Integer -> [Paren] +nthNestedParentheses = fasc4A_algorithm_U + +countNestedParentheses :: Int -> Integer +countNestedParentheses = countBinaryTrees + +-- | Generates all sequences of nested parentheses of length 2n. +-- Order is lexicographical (when right parentheses are considered +-- smaller then left ones). +-- Based on \"Algorithm P\" in Knuth, but less efficient because of +-- the \"idiomatic\" code. +fasc4A_algorithm_P :: Int -> [[Paren]] +fasc4A_algorithm_P 0 = [[]] +fasc4A_algorithm_P 1 = [[LeftParen,RightParen]] +fasc4A_algorithm_P n = unfold next ( start , [] ) where + start = concat $ replicate n [RightParen,LeftParen] -- already reversed! + + next :: ([Paren],[Paren]) -> ( [Paren] , Maybe ([Paren],[Paren]) ) + next ( (a:b:ls) , [] ) = next ( ls , b:a:[] ) + next ( lls@(l:ls) , rrs@(r:rs) ) = ( visit , new ) where + visit = reverse lls ++ rrs + new = + {- debug (reverse ls,l,r,rs) $ -} + case l of + RightParen -> Just ( ls , LeftParen:RightParen:rs ) + LeftParen -> + {- debug ("---",reverse ls,l,r,rs) $ -} + findj ( lls , [] ) ( reverse (RightParen:rs) , [] ) + next _ = error "fasc4A_algorithm_P: fatal error shouldn't happen" + + findj :: ([Paren],[Paren]) -> ([Paren],[Paren]) -> Maybe ([Paren],[Paren]) + findj ( [] , _ ) _ = Nothing + findj ( lls@(l:ls) , rs) ( xs , ys ) = + {- debug ((reverse ls,l,rs),(reverse xs,ys)) $ -} + case l of + LeftParen -> case xs of + (a:_:as) -> findj ( ls, RightParen:rs ) ( as , LeftParen:a:ys ) + _ -> findj ( lls, [] ) ( reverse rs ++ xs , ys) + RightParen -> Just ( reverse ys ++ xs ++ reverse (LeftParen:rs) ++ ls , [] ) + +-- | Generates a uniformly random sequence of nested parentheses of length 2n. +-- Based on \"Algorithm W\" in Knuth. +fasc4A_algorithm_W :: RandomGen g => Int -> g -> ([Paren],g) +fasc4A_algorithm_W n' rnd = worker (rnd,n,n,[]) where + n = fromIntegral n' :: Integer + -- the numbers we use are of order n^2, so for n >> 2^16 + -- on a 32 bit machine, we need big integers. + worker :: RandomGen g => (g,Integer,Integer,[Paren]) -> ([Paren],g) + worker (rnd,_,0,parens) = (parens,rnd) + worker (rnd,p,q,parens) = + if x<(q+1)*(q-p) + then worker (rnd' , p , q-1 , LeftParen :parens) + else worker (rnd' , p-1 , q , RightParen:parens) + where + (x,rnd') = randomR ( 0 , (q+p)*(q-p+1)-1 ) rnd + +-- | Nth sequence of nested parentheses of length 2n. +-- The order is the same as in 'fasc4A_algorithm_P'. +-- Based on \"Algorithm U\" in Knuth. +fasc4A_algorithm_U + :: Int -- ^ n + -> Integer -- ^ N; should satisfy 1 <= N <= C(n) + -> [Paren] +fasc4A_algorithm_U n' bign0 = reverse $ worker (bign0,c0,n,n,[]) where + n = fromIntegral n' :: Integer + c0 = foldl f 1 [2..n] + f c p = ((4*p-2)*c) `div` (p+1) + worker :: (Integer,Integer,Integer,Integer,[Paren]) -> [Paren] + worker (_ ,_,_,0,parens) = parens + worker (bign,c,p,q,parens) = + if bign <= c' + then worker (bign , c' , p , q-1 , RightParen:parens) + else worker (bign-c' , c-c' , p-1 , q , LeftParen :parens) + where + c' = ((q+1)*(q-p)*c) `div` ((q+p)*(q-p+1)) + +-------------------------------------------------------------------------------- +-- * Generating binary trees + +-- | Generates all binary trees with @n@ nodes. +-- At the moment just a synonym for 'binaryTreesNaive'. +binaryTrees :: Int -> [BinTree ()] +binaryTrees = binaryTreesNaive + +-- | # = Catalan(n) = \\frac { 1 } { n+1 } \\binom { 2n } { n }. +-- +-- This is also the counting function for forests and nested parentheses. +countBinaryTrees :: Int -> Integer +countBinaryTrees n = binomial (2*n) n `div` (1 + fromIntegral n) + +-- | Generates all binary trees with n nodes. The naive algorithm. +binaryTreesNaive :: Int -> [BinTree ()] +binaryTreesNaive 0 = [ leaf ] +binaryTreesNaive n = + [ Branch l r + | i <- [0..n-1] + , l <- binaryTreesNaive i + , r <- binaryTreesNaive (n-1-i) + ] + +-- | Generates an uniformly random binary tree, using 'fasc4A_algorithm_R'. +randomBinaryTree :: RandomGen g => Int -> g -> (BinTree (), g) +randomBinaryTree n rnd = (tree,rnd') where + (decorated,rnd') = fasc4A_algorithm_R n rnd + tree = fmap (const ()) $ forgetNodeDecorations decorated + +-- | Grows a uniformly random binary tree. +-- \"Algorithm R\" (Remy's procudere) in Knuth. +-- Nodes are decorated with odd numbers, leaves with even numbers (from the +-- set @[0..2n]@). Uses mutable arrays internally. +fasc4A_algorithm_R :: RandomGen g => Int -> g -> (BinTree' Int Int, g) +fasc4A_algorithm_R n0 rnd = res where + res = runST $ do + ar <- newArray (0,2*n0) 0 + rnd' <- worker rnd 1 ar + links <- Data.Array.Unsafe.unsafeFreeze ar + return (toTree links, rnd') + toTree links = f (links!0) where + f i = if odd i + then Branch' (f $ links!i) i (f $ links!(i+1)) + else Leaf' i + worker :: RandomGen g => g -> Int -> STUArray s Int Int -> ST s g + worker rnd n ar = do + if n > n0 + then return rnd + else do + writeArray ar (n2-b) n2 + lk <- readArray ar k + writeArray ar (n2-1+b) lk + writeArray ar k (n2-1) + worker rnd' (n+1) ar + where + n2 = n+n + (x,rnd') = randomR (0,4*n-3) rnd + (k,b) = x `divMod` 2 + +-------------------------------------------------------------------------------- +-- * ASCII drawing + +-- | Draws a binary tree in ASCII, ignoring node labels. +-- +-- Example: +-- +-- > autoTabulate RowMajor (Right 5) $ map asciiBinaryTree_ $ binaryTrees 4 +-- +asciiBinaryTree_ :: BinTree a -> ASCII +asciiBinaryTree_ = ASCII.asciiFromLines . fst . go where + + go :: BinTree a -> ([String],Int) + go (Leaf x) = ([],0) + go (Branch t1 t2) = ( new , j1+m ) where + (ls1,j1) = go t1 + (ls2,j2) = go t2 + w1 = blockWidth ls1 + w2 = blockWidth ls2 + m = max 1 $ (w1-j1+j2+2) `div` 2 + s = 2*m - (w1-j1+j2) + spaces = [replicate s ' '] + ls = hConcatLines [ ls1 , spaces , ls2 ] + top = [ replicate (j1+m-i) ' ' ++ "/" ++ replicate (2*(i-1)) ' ' ++ "\\" | i<-[1..m] ] + new = mkLinesUniformWidth $ vConcatLines [ top , ls ] + + blockWidth ls = case ls of + (l:_) -> length l + [] -> 0 + +instance DrawASCII (BinTree ()) where + ascii = asciiBinaryTree_ + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Trees/Binary.hs-boot b/Math/Combinat/Trees/Binary.hs-boot new file mode 100644 --- /dev/null +++ b/Math/Combinat/Trees/Binary.hs-boot @@ -0,0 +1,22 @@ + + +module Math.Combinat.Trees.Binary where + +-------------------------------------------------------------------------------- + +import Data.Tree ( Tree(..) , Forest(..) ) + +-------------------------------------------------------------------------------- + +-- | A binary tree with leaves decorated with type @a@. +data BinTree a + = Branch (BinTree a) (BinTree a) + | Leaf a + +-- | A binary tree with leaves and internal nodes decorated +-- with types @a@ and @b@, respectively. +data BinTree' a b + = Branch' (BinTree' a b) b (BinTree' a b) + | Leaf' a + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Trees/Graphviz.hs b/Math/Combinat/Trees/Graphviz.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Trees/Graphviz.hs @@ -0,0 +1,115 @@ + +-- | Creates graphviz @.dot@ files from trees. + +module Math.Combinat.Trees.Graphviz + ( Dot + , graphvizDotBinTree + , graphvizDotBinTree' + , graphvizDotForest + , graphvizDotTree + ) + where + +-------------------------------------------------------------------------------- + +import Data.Tree + +import Control.Applicative + +import {-# SOURCE #-} Math.Combinat.Trees.Binary ( BinTree(..) , BinTree'(..) ) +import {-# SOURCE #-} Math.Combinat.Trees.Nary ( addUniqueLabelsTree , addUniqueLabelsForest ) + +-------------------------------------------------------------------------------- + +type Dot = String + +digraphBracket :: String -> [String] -> String +digraphBracket name lines = + "digraph " ++ name ++ " {\n" ++ + concatMap (\xs -> " "++xs++"\n") lines + ++ "}\n" + +-------------------------------------------------------------------------------- + +graphvizDotBinTree :: Show a => String -> BinTree a -> Dot +graphvizDotBinTree graphname tree = + digraphBracket graphname $ binTreeDot' tree + +graphvizDotBinTree' :: (Show a, Show b) => String -> BinTree' a b -> Dot +graphvizDotBinTree' graphname tree = + digraphBracket graphname $ binTree'Dot' tree + +binTreeDot' :: Show a => BinTree a -> [String] +binTreeDot' tree = lines where + lines = worker (0::Int) "r" tree + name path = "node_"++path + worker depth path (Leaf x) = + [ name path ++ "[shape=box,label=\"" ++ show x ++ "\"" ++ "];" ] + worker depth path (Branch left right) + = [vertex,leftedge,rightedge] ++ + worker (depth+1) ('l':path) left ++ + worker (depth+1) ('r':path) right + where + vertex = name path ++ "[shape=circle,style=filled,height=0.25,label=\"\"];" + leftedge = name path ++ " -> " ++ name ('l':path) ++ "[tailport=sw];" + rightedge = name path ++ " -> " ++ name ('r':path) ++ "[tailport=se];" + +binTree'Dot' :: (Show a, Show b) => BinTree' a b -> [String] +binTree'Dot' tree = lines where + lines = worker (0::Int) "r" tree + name path = "node_"++path + worker depth path (Leaf' x) = + [ name path ++ "[shape=box,label=\"" ++ show x ++ "\"" ++ "];" ] + worker depth path (Branch' left y right) + = [vertex,leftedge,rightedge] ++ + worker (depth+1) ('l':path) left ++ + worker (depth+1) ('r':path) right + where + vertex = name path ++ "[shape=ellipse,label=\"" ++ show y ++ "\"];" + leftedge = name path ++ " -> " ++ name ('l':path) ++ "[tailport=sw];" + rightedge = name path ++ " -> " ++ name ('r':path) ++ "[tailport=se];" + +-------------------------------------------------------------------------------- + +-- | Generates graphviz @.dot@ file from a forest. The first argument tells whether +-- to make the individual trees clustered subgraphs; the second is the name of the +-- graph. +graphvizDotForest + :: Show a + => Bool -- ^ make the individual trees clustered subgraphs + -> Bool -- ^ reverse the direction of the arrows + -> String -- ^ name of the graph + -> Forest a + -> Dot +graphvizDotForest clustered revarrows graphname forest = digraphBracket graphname lines where + lines = concat $ zipWith cluster [(1::Int)..] (addUniqueLabelsForest forest) + name unique = "node_"++show unique + cluster j tree = let treelines = worker (0::Int) tree in case clustered of + False -> treelines + True -> ("subgraph cluster_"++show j++" {") : map (" "++) treelines ++ ["}"] + worker depth (Node (label,unique) subtrees) = vertex : edges ++ concatMap (worker (depth+1)) subtrees where + vertex = name unique ++ "[label=\"" ++ show label ++ "\"" ++ "];" + edges = map edge subtrees + edge (Node (_,unique') _) = if not revarrows + then name unique ++ " -> " ++ name unique' + else name unique' ++ " -> " ++ name unique + +-- | Generates graphviz @.dot@ file from a tree. The first argument is +-- the name of the graph. +graphvizDotTree + :: Show a + => Bool -- ^ reverse the direction of the arrow + -> String -- ^ name of the graph + -> Tree a + -> Dot +graphvizDotTree revarrows graphname tree = digraphBracket graphname lines where + lines = worker (0::Int) (addUniqueLabelsTree tree) + name unique = "node_"++show unique + worker depth (Node (label,unique) subtrees) = vertex : edges ++ concatMap (worker (depth+1)) subtrees where + vertex = name unique ++ "[label=\"" ++ show label ++ "\"" ++ "];" + edges = map edge subtrees + edge (Node (_,unique') _) = if not revarrows + then name unique ++ " -> " ++ name unique' + else name unique' ++ " -> " ++ name unique + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Trees/Nary.hs b/Math/Combinat/Trees/Nary.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Trees/Nary.hs @@ -0,0 +1,432 @@ + +-- | N-ary trees. + +{-# LANGUAGE FlexibleInstances #-} +{-# LANGUAGE TypeSynonymInstances #-} +module Math.Combinat.Trees.Nary + ( + -- * Types + module Data.Tree + , Tree(..) + -- * Regular trees + , ternaryTrees + , regularNaryTrees + , semiRegularTrees + , countTernaryTrees + , countRegularNaryTrees + -- * \"derivation trees\" + , derivTrees + -- * ASCII drawings + , asciiTreeVertical_ + , asciiTreeVertical + , asciiTreeVerticalLeavesOnly + -- * Graphviz drawing + , Dot + , graphvizDotTree + , graphvizDotForest + -- * Classifying nodes + , classifyTreeNode + , isTreeLeaf , isTreeNode + , isTreeLeaf_ , isTreeNode_ + , treeNodeNumberOfChildren + -- * Counting nodes + , countTreeNodes + , countTreeLeaves + , countTreeLabelsWith + , countTreeNodesWith + -- * Left and right spines + , leftSpine , leftSpine_ + , rightSpine , rightSpine_ + , leftSpineLength , rightSpineLength + -- * Unique labels + , addUniqueLabelsTree + , addUniqueLabelsForest + , addUniqueLabelsTree_ + , addUniqueLabelsForest_ + -- * Labelling by depth + , labelDepthTree + , labelDepthForest + , labelDepthTree_ + , labelDepthForest_ + -- * Labelling by number of children + , labelNChildrenTree + , labelNChildrenForest + , labelNChildrenTree_ + , labelNChildrenForest_ + + ) where + + +-------------------------------------------------------------------------------- + +import Data.List +import Data.Tree + +import Control.Applicative + +--import Control.Monad.State +import Control.Monad.Trans.State +import Data.Traversable (traverse) + +import Math.Combinat.Compositions (compositions) +import Math.Combinat.Numbers (binomial, factorial) +import Math.Combinat.Partitions.Multiset (partitionMultiset) +import Math.Combinat.Sets (listTensor) + +import Math.Combinat.Trees.Graphviz (Dot, graphvizDotForest, + graphvizDotTree) + +import Math.Combinat.ASCII as ASCII +import Math.Combinat.Classes +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- + +instance HasNumberOfNodes (Tree a) where + numberOfNodes = go where + go (Node label subforest) = if null subforest + then 0 + else 1 + sum' (map go subforest) + +instance HasNumberOfLeaves (Tree a) where + numberOfLeaves = go where + go (Node label subforest) = if null subforest + then 1 + else sum' (map go subforest) + +-------------------------------------------------------------------------------- + +-- | @regularNaryTrees d n@ returns the list of (rooted) trees on @n@ nodes where each +-- node has exactly @d@ children. Note that the leaves do not count in @n@. +-- Naive algorithm. +regularNaryTrees + :: Int -- ^ degree = number of children of each node + -> Int -- ^ number of nodes + -> [Tree ()] +regularNaryTrees d = go where + go 0 = [ Node () [] ] + go n = [ Node () cs + | is <- compositions d (n-1) + , cs <- listTensor [ go i | i<-is ] + ] + +-- | Ternary trees on @n@ nodes (synonym for @regularNaryTrees 3@) +ternaryTrees :: Int -> [Tree ()] +ternaryTrees = regularNaryTrees 3 + +-- | We have +-- +-- > length (regularNaryTrees d n) == countRegularNaryTrees d n == \frac {1} {(d-1)n+1} \binom {dn} {n} +-- +countRegularNaryTrees :: (Integral a, Integral b) => a -> b -> Integer +countRegularNaryTrees d n = binomial (dd*nn) nn `div` ((dd-1)*nn+1) where + dd = fromIntegral d :: Integer + nn = fromIntegral n :: Integer + +-- | @\# = \\frac {1} {(2n+1} \\binom {3n} {n}@ +countTernaryTrees :: Integral a => a -> Integer +countTernaryTrees = countRegularNaryTrees (3::Int) + +-------------------------------------------------------------------------------- + +-- | All trees on @n@ nodes where the number of children of all nodes is +-- in element of the given set. Example: +-- +-- > autoTabulate RowMajor (Right 5) $ map asciiTreeVertical +-- > $ map labelNChildrenTree_ +-- > $ semiRegularTrees [2,3] 2 +-- > +-- > [ length $ semiRegularTrees [2,3] n | n<-[0..] ] == [1,2,10,66,498,4066,34970,312066,2862562,26824386,...] +-- +-- The latter sequence is A027307 in OEIS: +-- +-- Remark: clearly, we have +-- +-- > semiRegularTrees [d] n == regularNaryTrees d n +-- +-- +semiRegularTrees + :: [Int] -- ^ set of allowed number of children + -> Int -- ^ number of nodes + -> [Tree ()] +semiRegularTrees [] n = if n==0 then [Node () []] else [] +semiRegularTrees dset_ n = + if head dset >=1 + then go n + else error "semiRegularTrees: expecting a list of positive integers" + where + dset = map head $ group $ sort $ dset_ + + go 0 = [ Node () [] ] + go n = [ Node () cs + | d <- dset + , is <- compositions d (n-1) + , cs <- listTensor [ go i | i<-is ] + ] + +{- + +NOTES: + +A006318 = [ length $ semiRegularTrees [1,2] n | n<-[0..] ] == [1,2,6,22,90,394,1806,8558,41586,206098,1037718.. ] +?? = [ length $ semiRegularTrees [1,3] n | n<-[0..] ] == [1,2,8,44,280,1936,14128,107088,834912,6652608 .. ] +?? = [ length $ semiRegularTrees [1,4] n | n<-[0..] ] == [1,2,10,74,642,6082,60970,635818,6826690 + +A027307 = [ length $ semiRegularTrees [2,3] n | n<-[0..] ] == [1,2,10,66,498,4066,34970,312066,2862562,26824386,...] +A219534 = [ length $ semiRegularTrees [2,4] n | n<-[0..] ] == [1,2,12,100,968,10208,113792,1318832 ..] +?? = [ length $ semiRegularTrees [2,5] n | n<-[0..] ] == [1,2,14,142,1690,21994,303126,4348102 ..] + +A144097 = [ length $ semiRegularTrees [3,4] n | n<-[0..] ] == [1,2,14,134,1482,17818,226214,2984206,40503890..] + +A107708 = [ length $ semiRegularTrees [1,2,3] n | n<-[0..] ] == [1,3,18,144,1323,13176,138348,1507977 .. ] +?? = [ length $ semiRegularTrees [1,2,3,4] n | n<-[0..] ] == [1,4,40,560,9120,161856,3036800,59242240 .. ] + +-} + +-------------------------------------------------------------------------------- + +-- | Vertical ASCII drawing of a tree, without labels. Example: +-- +-- > autoTabulate RowMajor (Right 5) $ map asciiTreeVertical_ $ regularNaryTrees 2 4 +-- +-- Nodes are denoted by @\@@, leaves by @*@. +-- +asciiTreeVertical_ :: Tree a -> ASCII +asciiTreeVertical_ tree = ASCII.asciiFromLines (go tree) where + go :: Tree b -> [String] + go (Node _ cs) = case cs of + [] -> ["-*"] + _ -> concat $ mapWithFirstLast f $ map go cs + + f :: Bool -> Bool -> [String] -> [String] + f bf bl (l:ls) = let indent = if bl then " " else "| " + gap = if bl then [] else ["| "] + branch = if bl && not bf + then "\\-" + else if bf then "@-" + else "+-" + in (branch++l) : map (indent++) ls ++ gap + +instance DrawASCII (Tree ()) where + ascii = asciiTreeVertical_ + +-- | Prints all labels. Example: +-- +-- > asciiTreeVertical $ addUniqueLabelsTree_ $ (regularNaryTrees 3 9) !! 666 +-- +-- Nodes are denoted by @(label)@, leaves by @label@. +-- +asciiTreeVertical :: Show a => Tree a -> ASCII +asciiTreeVertical tree = ASCII.asciiFromLines (go tree) where + go :: Show b => Tree b -> [String] + go (Node x cs) = case cs of + [] -> ["-- " ++ show x] + _ -> concat $ mapWithFirstLast (f (show x)) $ map go cs + + f :: String -> Bool -> Bool -> [String] -> [String] + f label bf bl (l:ls) = + let spaces = (map (const ' ') label ) + dashes = (map (const '-') spaces ) + indent = if bl then " " ++spaces++" " else " |" ++ spaces ++ " " + gap = if bl then [] else [" |" ++ spaces ++ " "] + branch = if bl && not bf + then " \\"++dashes++"--" + else if bf + then "-(" ++ label ++ ")-" + else " +" ++ dashes ++ "--" + in (branch++l) : map (indent++) ls ++ gap + +-- | Prints the labels for the leaves, but not for the nodes. +asciiTreeVerticalLeavesOnly :: Show a => Tree a -> ASCII +asciiTreeVerticalLeavesOnly tree = ASCII.asciiFromLines (go tree) where + go :: Show b => Tree b -> [String] + go (Node x cs) = case cs of + [] -> ["- " ++ show x] + _ -> concat $ mapWithFirstLast f $ map go cs + + f :: Bool -> Bool -> [String] -> [String] + f bf bl (l:ls) = let indent = if bl then " " else "| " + gap = if bl then [] else ["| "] + branch = if bl && not bf + then "\\-" + else if bf then "@-" + else "+-" + in (branch++l) : map (indent++) ls ++ gap + +-------------------------------------------------------------------------------- + +-- | The leftmost spine (the second element of the pair is the leaf node) +leftSpine :: Tree a -> ([a],a) +leftSpine = go where + go (Node x cs) = case cs of + [] -> ([],x) + _ -> let (xs,y) = go (head cs) in (x:xs,y) + +rightSpine :: Tree a -> ([a],a) +rightSpine = go where + go (Node x cs) = case cs of + [] -> ([],x) + _ -> let (xs,y) = go (last cs) in (x:xs,y) + +-- | The leftmost spine without the leaf node +leftSpine_ :: Tree a -> [a] +leftSpine_ = go where + go (Node x cs) = case cs of + [] -> [] + _ -> x : go (head cs) + +rightSpine_ :: Tree a -> [a] +rightSpine_ = go where + go (Node x cs) = case cs of + [] -> [] + _ -> x : go (last cs) + +-- | The length (number of edges) on the left spine +-- +-- > leftSpineLength tree == length (leftSpine_ tree) +-- +leftSpineLength :: Tree a -> Int +leftSpineLength = go 0 where + go n (Node x cs) = case cs of + [] -> n + _ -> go (n+1) (head cs) + +rightSpineLength :: Tree a -> Int +rightSpineLength = go 0 where + go n (Node x cs) = case cs of + [] -> n + _ -> go (n+1) (last cs) + +-------------------------------------------------------------------------------- + +-- | 'Left' is leaf, 'Right' is node +classifyTreeNode :: Tree a -> Either a a +classifyTreeNode (Node x cs) = case cs of { [] -> Left x ; _ -> Right x } + +isTreeLeaf :: Tree a -> Maybe a +isTreeLeaf (Node x cs) = case cs of { [] -> Just x ; _ -> Nothing } + +isTreeNode :: Tree a -> Maybe a +isTreeNode (Node x cs) = case cs of { [] -> Nothing ; _ -> Just x } + +isTreeLeaf_ :: Tree a -> Bool +isTreeLeaf_ (Node x cs) = case cs of { [] -> True ; _ -> False } + +isTreeNode_ :: Tree a -> Bool +isTreeNode_ (Node x cs) = case cs of { [] -> False ; _ -> True } + +treeNodeNumberOfChildren :: Tree a -> Int +treeNodeNumberOfChildren (Node _ cs) = length cs + +-------------------------------------------------------------------------------- +-- counting + +countTreeNodes :: Tree a -> Int +countTreeNodes = go where + go (Node x cs) = case cs of + [] -> 0 + _ -> 1 + sum (map go cs) + +countTreeLeaves :: Tree a -> Int +countTreeLeaves = go where + go (Node x cs) = case cs of + [] -> 1 + _ -> sum (map go cs) + +countTreeLabelsWith :: (a -> Bool) -> Tree a -> Int +countTreeLabelsWith f = go where + go (Node label cs) = (if f label then 1 else 0) + sum (map go cs) + +countTreeNodesWith :: (Tree a -> Bool) -> Tree a -> Int +countTreeNodesWith f = go where + go node@(Node _ cs) = (if f node then 1 else 0) + sum (map go cs) + +-------------------------------------------------------------------------------- + +-- | Adds unique labels to the nodes (including leaves) of a 'Tree'. +addUniqueLabelsTree :: Tree a -> Tree (a,Int) +addUniqueLabelsTree tree = head (addUniqueLabelsForest [tree]) + +-- | Adds unique labels to the nodes (including leaves) of a 'Forest' +addUniqueLabelsForest :: Forest a -> Forest (a,Int) +addUniqueLabelsForest forest = evalState (mapM globalAction forest) 1 where + globalAction tree = + unwrapMonad $ traverse localAction tree + localAction x = WrapMonad $ do + i <- get + put (i+1) + return (x,i) + +addUniqueLabelsTree_ :: Tree a -> Tree Int +addUniqueLabelsTree_ = fmap snd . addUniqueLabelsTree + +addUniqueLabelsForest_ :: Forest a -> Forest Int +addUniqueLabelsForest_ = map (fmap snd) . addUniqueLabelsForest + +-------------------------------------------------------------------------------- + +-- | Attaches the depth to each node. The depth of the root is 0. +labelDepthTree :: Tree a -> Tree (a,Int) +labelDepthTree tree = worker 0 tree where + worker depth (Node label subtrees) = Node (label,depth) (map (worker (depth+1)) subtrees) + +labelDepthForest :: Forest a -> Forest (a,Int) +labelDepthForest forest = map labelDepthTree forest + +labelDepthTree_ :: Tree a -> Tree Int +labelDepthTree_ = fmap snd . labelDepthTree + +labelDepthForest_ :: Forest a -> Forest Int +labelDepthForest_ = map (fmap snd) . labelDepthForest + +-------------------------------------------------------------------------------- + +-- | Attaches the number of children to each node. +labelNChildrenTree :: Tree a -> Tree (a,Int) +labelNChildrenTree (Node x subforest) = + Node (x, length subforest) (map labelNChildrenTree subforest) + +labelNChildrenForest :: Forest a -> Forest (a,Int) +labelNChildrenForest forest = map labelNChildrenTree forest + +labelNChildrenTree_ :: Tree a -> Tree Int +labelNChildrenTree_ = fmap snd . labelNChildrenTree + +labelNChildrenForest_ :: Forest a -> Forest Int +labelNChildrenForest_ = map (fmap snd) . labelNChildrenForest + +-------------------------------------------------------------------------------- + +-- | Computes the set of equivalence classes of rooted trees (in the +-- sense that the leaves of a node are /unordered/) +-- with @n = length ks@ leaves where the set of heights of +-- the leaves matches the given set of numbers. +-- The height is defined as the number of /edges/ from the leaf to the root. +-- +-- TODO: better name? +derivTrees :: [Int] -> [Tree ()] +derivTrees xs = derivTrees' (map (+1) xs) + +derivTrees' :: [Int] -> [Tree ()] +derivTrees' [] = [] +derivTrees' [n] = + if n>=1 + then [unfoldTree f 1] + else [] + where + f k = if k0) ks) + then + [ Node () sub + | part <- parts + , let subtrees = map g part + , sub <- listTensor subtrees + ] + else [] + where + parts = partitionMultiset ks + g xs = derivTrees' (map (\x->x-1) xs) + +-------------------------------------------------------------------------------- + diff --git a/Math/Combinat/Trees/Nary.hs-boot b/Math/Combinat/Trees/Nary.hs-boot new file mode 100644 --- /dev/null +++ b/Math/Combinat/Trees/Nary.hs-boot @@ -0,0 +1,16 @@ + +module Math.Combinat.Trees.Nary where + +-------------------------------------------------------------------------------- + +import Data.Tree + +-------------------------------------------------------------------------------- + +addUniqueLabelsTree :: Tree a -> Tree (a,Int) +addUniqueLabelsForest :: Forest a -> Forest (a,Int) + +addUniqueLabelsTree_ :: Tree a -> Tree Int +addUniqueLabelsForest_ :: Forest a -> Forest Int + +-------------------------------------------------------------------------------- diff --git a/Math/Combinat/Tuples.hs b/Math/Combinat/Tuples.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/Tuples.hs @@ -0,0 +1,61 @@ + +-- | Tuples. + +module Math.Combinat.Tuples where + +import Math.Combinat.Helper + +------------------------------------------------------- +-- Tuples + +-- | \"Tuples\" fitting into a give shape. The order is lexicographic, that is, +-- +-- > sort ts == ts where ts = tuples' shape +-- +-- Example: +-- +-- > tuples' [2,3] = +-- > [[0,0],[0,1],[0,2],[0,3],[1,0],[1,1],[1,2],[1,3],[2,0],[2,1],[2,2],[2,3]] +-- +tuples' :: [Int] -> [[Int]] +tuples' [] = [[]] +tuples' (s:ss) = [ x:xs | x <- [0..s] , xs <- tuples' ss ] + +-- | positive \"tuples\" fitting into a give shape. +tuples1' :: [Int] -> [[Int]] +tuples1' [] = [[]] +tuples1' (s:ss) = [ x:xs | x <- [1..s] , xs <- tuples1' ss ] + +-- | # = \\prod_i (m_i + 1) +countTuples' :: [Int] -> Integer +countTuples' shape = product $ map f shape where + f k = 1 + fromIntegral k + +-- | # = \\prod_i m_i +countTuples1' :: [Int] -> Integer +countTuples1' shape = product $ map fromIntegral shape + +tuples + :: Int -- ^ length (width) + -> Int -- ^ maximum (height) + -> [[Int]] +tuples len k = tuples' (replicate len k) + +tuples1 + :: Int -- ^ length (width) + -> Int -- ^ maximum (height) + -> [[Int]] +tuples1 len k = tuples1' (replicate len k) + +-- | # = (m+1) ^ len +countTuples :: Int -> Int -> Integer +countTuples len k = (1 + fromIntegral k) ^ len + +-- | # = m ^ len +countTuples1 :: Int -> Int -> Integer +countTuples1 len k = fromIntegral k ^ len + +binaryTuples :: Int -> [[Bool]] +binaryTuples len = map (map intToBool) (tuples len 1) + +------------------------------------------------------- diff --git a/Math/Combinat/TypeLevel.hs b/Math/Combinat/TypeLevel.hs new file mode 100644 --- /dev/null +++ b/Math/Combinat/TypeLevel.hs @@ -0,0 +1,117 @@ + +-- | Type-level hackery. +-- +-- This module is used for groups whose parameters are encoded as type-level natural numbers, +-- for example finite cyclic groups, free groups, symmetric groups and braid groups. +-- + +{-# LANGUAGE PolyKinds, DataKinds, KindSignatures, ScopedTypeVariables, + ExistentialQuantification, Rank2Types #-} + +module Math.Combinat.TypeLevel + ( -- * Proxy + Proxy(..) + , proxyUndef + , proxyOf + , proxyOf1 + , proxyOf2 + , asProxyTypeOf -- defined in Data.Proxy + , asProxyTypeOf1 + -- * Type-level naturals as type arguments + , typeArg + , iTypeArg + -- * Hiding the type parameter + , Some (..) + , withSome , withSomeM + , selectSome , selectSomeM + , withSelected , withSelectedM + ) + where + +-------------------------------------------------------------------------------- + +import Data.Proxy ( Proxy(..) , asProxyTypeOf ) +import GHC.TypeLits + +import Math.Combinat.Numbers.Primes ( isProbablyPrime ) + +-------------------------------------------------------------------------------- +-- * Proxy + +proxyUndef :: Proxy a -> a +proxyUndef _ = error "proxyUndef" + +proxyOf :: a -> Proxy a +proxyOf _ = Proxy + +proxyOf1 :: f a -> Proxy a +proxyOf1 _ = Proxy + +proxyOf2 :: g (f a) -> Proxy a +proxyOf2 _ = Proxy + +asProxyTypeOf1 :: f a -> Proxy a -> f a +asProxyTypeOf1 y _ = y + +-------------------------------------------------------------------------------- +-- * Type-level naturals as type arguments + +typeArg :: KnownNat n => f (n :: Nat) -> Integer +typeArg = natVal . proxyOf1 + +iTypeArg :: KnownNat n => f (n :: Nat) -> Int +iTypeArg = fromIntegral . typeArg + +-------------------------------------------------------------------------------- +-- * Hiding the type parameter + +-- | Hide the type parameter of a functor. Example: @Some Braid@ +data Some f = forall n. KnownNat n => Some (f n) + +-- | Uses the value inside a 'Some' +withSome :: Some f -> (forall n. KnownNat n => f n -> a) -> a +withSome some polyFun = case some of { Some stuff -> polyFun stuff } + +-- | Monadic version of 'withSome' +withSomeM :: Monad m => Some f -> (forall n. KnownNat n => f n -> m a) -> m a +withSomeM some polyAct = case some of { Some stuff -> polyAct stuff } + +-- | Given a polymorphic value, we select at run time the +-- one specified by the second argument +selectSome :: Integral int => (forall n. KnownNat n => f n) -> int -> Some f +selectSome poly n = case someNatVal (fromIntegral n :: Integer) of + Nothing -> error "selectSome: not a natural number" + Just snat -> case snat of + SomeNat pxy -> Some (asProxyTypeOf1 poly pxy) + +-- | Monadic version of 'selectSome' +selectSomeM :: forall m f int. (Integral int, Monad m) => (forall n. KnownNat n => m (f n)) -> int -> m (Some f) +selectSomeM runPoly n = case someNatVal (fromIntegral n :: Integer) of + Nothing -> error "selectSomeM: not a natural number" + Just snat -> case snat of + SomeNat pxy -> do + poly <- runPoly + return $ Some (asProxyTypeOf1 poly pxy) + +-- | Combination of 'selectSome' and 'withSome': we make a temporary structure +-- of the given size, but we immediately consume it. +withSelected + :: Integral int + => (forall n. KnownNat n => f n -> a) + -> (forall n. KnownNat n => f n) + -> int + -> a +withSelected f x n = withSome (selectSome x n) f + +-- | (Half-)monadic version of 'withSelected' +withSelectedM + :: forall m f int a. (Integral int, Monad m) + => (forall n. KnownNat n => f n -> a) + -> (forall n. KnownNat n => m (f n)) + -> int + -> m a +withSelectedM f mx n = do + s <- selectSomeM mx n + return (withSome s f) + +-------------------------------------------------------------------------------- diff --git a/Setup.lhs b/Setup.lhs new file mode 100644 --- /dev/null +++ b/Setup.lhs @@ -0,0 +1,3 @@ +#! /usr/bin/env runhaskell +> import Distribution.Simple +> main = defaultMain diff --git a/combinat-compat.cabal b/combinat-compat.cabal new file mode 100644 --- /dev/null +++ b/combinat-compat.cabal @@ -0,0 +1,102 @@ +cabal-version: 1.18 +name: combinat-compat +version: 0.2.8.2 +license: BSD3 +license-file: LICENSE +copyright: (c) 2008-2016 Balazs Komuves +maintainer: bkomuves (plus) hackage (at) gmail (dot) com +author: Balazs Komuves +stability: Experimental +tested-with: ghc ==8.4.1 +homepage: http://code.haskell.org/~bkomuves/ +synopsis: Generate and manipulate various combinatorial objects. +description: + A collection of functions to generate, count, manipulate + and visualize all kinds of combinatorial objects like + partitions, compositions, trees, permutations, braids, + Young tableaux, and so on. + Forked from the [combinat](http://hackage.haskell.org/package/combinat) package. +category: Math +build-type: Simple +extra-source-files: + svg/*.svg + svg/src/gen_figures.hs +extra-doc-files: svg/*.svg + +library + exposed-modules: + Math.Combinat + Math.Combinat.Classes + Math.Combinat.Numbers + Math.Combinat.Numbers.Series + Math.Combinat.Numbers.Primes + Math.Combinat.Sign + Math.Combinat.Sets + Math.Combinat.Tuples + Math.Combinat.Compositions + Math.Combinat.Groups.Thompson.F + Math.Combinat.Groups.Free + Math.Combinat.Groups.Braid + Math.Combinat.Groups.Braid.NF + Math.Combinat.Partitions + Math.Combinat.Partitions.Integer + Math.Combinat.Partitions.Skew + Math.Combinat.Partitions.Set + Math.Combinat.Partitions.NonCrossing + Math.Combinat.Partitions.Plane + Math.Combinat.Partitions.Multiset + Math.Combinat.Partitions.Vector + Math.Combinat.Permutations + Math.Combinat.Tableaux + Math.Combinat.Tableaux.Skew + Math.Combinat.Tableaux.GelfandTsetlin + Math.Combinat.Tableaux.GelfandTsetlin.Cone + Math.Combinat.Tableaux.LittlewoodRichardson + Math.Combinat.Trees + Math.Combinat.Trees.Binary + Math.Combinat.Trees.Nary + Math.Combinat.Trees.Graphviz + Math.Combinat.LatticePaths + Math.Combinat.ASCII + Math.Combinat.Helper + Math.Combinat.TypeLevel + hs-source-dirs: . + default-language: Haskell2010 + default-extensions: CPP BangPatterns + other-extensions: MultiParamTypeClasses ScopedTypeVariables + GeneralizedNewtypeDeriving DataKinds KindSignatures + ghc-options: -fwarn-tabs -fno-warn-unused-matches + -fno-warn-name-shadowing -fno-warn-unused-imports + build-depends: + base >=4.11 && <5, + array >=0.5, + containers -any, + random -any, + transformers -any + +test-suite combinat-tests + type: exitcode-stdio-1.0 + main-is: TestSuite.hs + hs-source-dirs: test + other-modules: + Tests.Braid + Tests.Common + Tests.LatticePaths + Tests.Permutations + Tests.Series + Tests.SkewTableaux + Tests.Thompson + Tests.Partitions.Integer + Tests.Partitions.Skew + default-language: Haskell2010 + default-extensions: CPP BangPatterns + build-depends: + base >=4 && <5, + array >=0.5, + containers -any, + random -any, + transformers -any, + combinat-compat -any, + QuickCheck >=2, + test-framework -any, + test-framework-quickcheck2 -any diff --git a/svg/bintrees.svg b/svg/bintrees.svg new file mode 100644 --- /dev/null +++ b/svg/bintrees.svg @@ -0,0 +1,4 @@ + + + diff --git a/svg/dyck_path.svg b/svg/dyck_path.svg new file mode 100644 --- /dev/null +++ b/svg/dyck_path.svg @@ -0,0 +1,3 @@ + + diff --git a/svg/ferrers.svg b/svg/ferrers.svg new file mode 100644 --- /dev/null +++ b/svg/ferrers.svg @@ -0,0 +1,4 @@ + + + diff --git a/svg/noncrossing.svg b/svg/noncrossing.svg new file mode 100644 --- /dev/null +++ b/svg/noncrossing.svg @@ -0,0 +1,4 @@ + + +987654321 diff --git a/svg/plane_partition.svg b/svg/plane_partition.svg new file mode 100644 --- /dev/null +++ b/svg/plane_partition.svg @@ -0,0 +1,4 @@ + + + diff --git a/svg/skew3.svg b/svg/skew3.svg new file mode 100644 --- /dev/null +++ b/svg/skew3.svg @@ -0,0 +1,3 @@ + + diff --git a/svg/skew_tableau.svg b/svg/skew_tableau.svg new file mode 100644 --- /dev/null +++ b/svg/skew_tableau.svg @@ -0,0 +1,3 @@ + +7512122111111 diff --git a/svg/src/gen_figures.hs b/svg/src/gen_figures.hs new file mode 100644 --- /dev/null +++ b/svg/src/gen_figures.hs @@ -0,0 +1,81 @@ + +-- | A script to generate the SVG figures in the documentation. +-- We use the @combinat-diagrams@ library for that. + +module Main where + +-------------------------------------------------------------------------------- + +import Math.Combinat.Partitions.Integer +import Math.Combinat.Partitions.Plane +import Math.Combinat.Partitions.NonCrossing +import Math.Combinat.Partitions.Skew +import Math.Combinat.Tableaux +import Math.Combinat.Tableaux.Skew +import Math.Combinat.LatticePaths +import Math.Combinat.Trees.Binary + +import Math.Combinat.Diagrams.Partitions.Integer +import Math.Combinat.Diagrams.Partitions.Plane +import Math.Combinat.Diagrams.Partitions.NonCrossing +import Math.Combinat.Diagrams.Partitions.Skew +import Math.Combinat.Diagrams.Tableaux +import Math.Combinat.Diagrams.Tableaux.Skew +import Math.Combinat.Diagrams.LatticePaths +import Math.Combinat.Diagrams.Trees.Binary + +import Diagrams.Core +import Diagrams.Prelude +import Diagrams.Backend.SVG + +-------------------------------------------------------------------------------- + +export fpath size what = renderSVG fpath size $ pad 1.10 what + +vcatSep = vcat' (with & sep .~ 1) +hcatSep = hcat' (with & sep .~ 1) + +boxSep m xs = pad 1.05 $ vcatSep $ map hcatSep $ yys where + yys = go xs where + go [] = [] + go zs = take m zs : go (drop m zs) + +padding fac diag = pad fac $ centerXY diag +margin siz diag = hcat [ strutX siz , vcat [ strutY siz , centerXY diag , strutY siz ] , strutX siz ] + +-------------------------------------------------------------------------------- + +main = do + + export "plane_partition.svg" (mkWidth 320) $ margin 0.05 $ drawPlanePartition3D $ + PlanePart [[5,4,3,3,1],[4,4,2,1],[3,2],[2,1],[1],[1]] + + export "noncrossing.svg" (mkWidth 256) $ padding 1.10 $ drawNonCrossingCircleDiagram' orange True $ + NonCrossing [[3],[5,4,2],[7,6,1],[9,8]] + + export "young_tableau.svg" (mkWidth 256) $ margin 0.05 $ drawTableau $ + [ [ 1 , 3 , 4 , 6 , 7 ] + , [ 2 , 5 , 8 ,10 ] + , [ 9 ] + ] + + let u = UpStep + d = DownStep + path = [ u,u,d,u,u,u,d,u,d,d,u,d,u,u,u,d,d,d,d,d,u,d,u,u,d,d ] + export "dyck_path.svg" (mkWidth 500) $ margin 0.05 $ drawLatticePath $ path + -- print (pathHeight path, pathNumberOfZeroTouches path, pathNumberOfPeaks path) + + export "ferrers.svg" (mkWidth 256) $ margin 0.05 $ drawFerrersDiagram' EnglishNotation red True $ + Partition [8,6,3,3,1] + + export "bintrees.svg" (mkWidth 750) $ boxSep 7 $ map drawBinTree_ (binaryTrees 4) + + let skew = mkSkewPartition (Partition [9,7,3,2,2,1] , Partition [5,3,2,1]) + -- export "skew.svg" (mkWidth 256) $ margin 0.05 $ drawSkewFerrersDiagram skew + -- export "skew2.svg" (mkWidth 256) $ margin 0.05 $ drawSkewFerrersDiagram' EnglishNotation green True (True,True) skew + export "skew3.svg" (mkWidth 256) $ margin 0.05 $ drawSkewPartitionBoxes EnglishNotation skew + + let skewtableau = (semiStandardSkewTableaux 7 skew) !! 123 + export "skew_tableau.svg" (mkWidth 320) $ margin 0.05 $ drawSkewTableau' EnglishNotation blue True skewtableau + +-------------------------------------------------------------------------------- diff --git a/svg/young_tableau.svg b/svg/young_tableau.svg new file mode 100644 --- /dev/null +++ b/svg/young_tableau.svg @@ -0,0 +1,4 @@ + + +91085276431 diff --git a/test/TestSuite.hs b/test/TestSuite.hs new file mode 100644 --- /dev/null +++ b/test/TestSuite.hs @@ -0,0 +1,41 @@ + +module Main where + +-------------------------------------------------------------------------------- + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 + +import Tests.Permutations ( testgroup_Permutations ) +import Tests.Partitions.Integer ( testgroup_IntegerPartitions ) +import Tests.Partitions.Skew ( testgroup_SkewPartitions ) +import Tests.Braid ( testgroup_Braid + , testgroup_Braid_NF ) +import Tests.Series ( testgroup_PowerSeries ) +import Tests.SkewTableaux ( testgroup_SkewTableaux ) +import Tests.Thompson ( testgroup_ThompsonF ) +import Tests.LatticePaths ( testgroup_LatticePaths ) + +-------------------------------------------------------------------------------- + +main :: IO () +main = defaultMain tests + +tests :: [Test] +tests = + [ testgroup_Permutations + , testGroup "Partitions" + [ testgroup_IntegerPartitions + , testgroup_SkewPartitions + ] + , testgroup_SkewTableaux + , testgroup_ThompsonF + , testgroup_LatticePaths + , testGroup "Braids" + [ testgroup_Braid + , testgroup_Braid_NF + ] + , testgroup_PowerSeries + ] + +-------------------------------------------------------------------------------- diff --git a/test/Tests/Braid.hs b/test/Tests/Braid.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Braid.hs @@ -0,0 +1,278 @@ + +-- | Tests for braids. + +{-# LANGUAGE + CPP, BangPatterns, + ScopedTypeVariables, ExistentialQuantification, + DataKinds, KindSignatures, Rank2Types, + TypeOperators, TypeFamilies, + StandaloneDeriving #-} + +module Tests.Braid where + +-------------------------------------------------------------------------------- + +import Math.Combinat.Groups.Braid +import Math.Combinat.Groups.Braid.NF + +import Tests.Permutations () -- arbitrary instance +import Tests.Common + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import Test.QuickCheck.Gen + +import Data.Proxy +import GHC.TypeLits + +import Control.Monad + +import Data.List ( mapAccumL , foldl' ) + +import Data.Array.Unboxed +import Data.Array.ST +import Data.Array.IArray +import Data.Array.MArray +import Data.Array.Unsafe +import Data.Array.Base + +import Control.Monad.ST + +import System.Random + +import Math.Combinat.Sign +import Math.Combinat.Helper +import Math.Combinat.TypeLevel +import Math.Combinat.Numbers.Series + +import Math.Combinat.Permutations ( Permutation(..) ) +import qualified Math.Combinat.Permutations as P + +-------------------------------------------------------------------------------- +-- * Types and instances + +maxBraidWordLen :: Int +maxBraidWordLen = 600 + +maxStrands :: Int +maxStrands = 18 -- normal forms are very slow for large ones + +shrinkBraid :: KnownNat n => Braid n -> [Braid n] +shrinkBraid (Braid gens) = map Braid list where + len = length gens + list = [ take i gens ++ drop (i+1) gens | i<-[0..len-1] ] + +-- someRndBraid :: Int -> (forall (n :: Nat). KnownNat n => g -> (Braid n, g)) -> g -> (SomeBraid, g) +-- someRndBraid n f = \g -> let (x,g') = f g in (someBraid n x, g') + +-- | equality as /braid words/ +(=:=) :: Braid n -> Braid n -> Bool +(=:=) (Braid gens1) (Braid gens2) = (gens1 == gens2) + +data UnreducedBraid = forall n. KnownNat n => Unreduced (Braid n) +data ReducedBraid = forall n. KnownNat n => Reduced (Braid n) +data PositiveBraid = forall n. KnownNat n => PositiveB (Braid n) +data PerturbedBraid = forall n. KnownNat n => Perturbed (Braid n) (Braid n) +data PermutationBraid = forall n. KnownNat n => PermBraid Permutation (Braid n) +data TwoBraids = forall n. KnownNat n => TwoBraids (Braid n) (Braid n) + +deriving instance Show UnreducedBraid +deriving instance Show ReducedBraid +deriving instance Show PositiveBraid +deriving instance Show PerturbedBraid +deriving instance Show PermutationBraid +deriving instance Show TwoBraids + +instance KnownNat n => Random (Braid n) where + randomR _ = random + random g0 = (b, g2) where + n = numberOfStrands b + (l,g1) = randomR (0,maxBraidWordLen) g0 + (b,g2) = randomBraidWord l g1 + +instance Random UnreducedBraid where + randomR _ = random + random = runRand $ do + n <- randChoose (2,maxStrands) + l <- randChoose (0,maxBraidWordLen) + withSelectedM Unreduced (rand $ randomBraidWord l) n + +instance Random PositiveBraid where + randomR _ = random + random = runRand $ do + n <- randChoose (2,maxStrands) + l <- randChoose (0,maxBraidWordLen) + withSelectedM PositiveB (rand $ randomPositiveBraidWord l) n + +instance Random PerturbedBraid where + randomR _ = random + random = runRand $ do + Unreduced b <- rand random + k <- randChoose (20,1000) + c <- rand $ randomPerturbBraidWord k b + return (Perturbed b c) + +instance KnownNat n => Arbitrary (Braid n) where + arbitrary = choose_ + shrink = shrinkBraid + +instance Arbitrary UnreducedBraid where + arbitrary = choose_ + shrink (Unreduced b) = map Unreduced (shrinkBraid b) + +instance Arbitrary PositiveBraid where + arbitrary = choose_ + shrink (PositiveB b) = map PositiveB (shrinkBraid b) + +instance Arbitrary ReducedBraid where + arbitrary = do + Unreduced braid <- arbitrary + return $ Reduced $ freeReduceBraidWord braid + +instance Arbitrary PerturbedBraid where + arbitrary = choose_ + shrink _ = [] + +instance Arbitrary TwoBraids where + shrink _ = [] + arbitrary = do + n <- choose (2::Int, maxStrands) + let snat = case someNatVal (fromIntegral n :: Integer) of + Just sn -> sn + Nothing -> error "TwoBraids/arbitrary: shouldn't happen" + case snat of + SomeNat pxy -> do + (braid1,braid2) <- choosePair_ + return $ TwoBraids (asProxyTypeOf1 braid1 pxy) (asProxyTypeOf1 braid2 pxy) + +mkPermBraid :: Permutation -> PermutationBraid +mkPermBraid perm = + case snat of + SomeNat pxy -> PermBraid perm (asProxyTypeOf1 (permutationBraid perm) pxy) + where + n = P.permutationSize perm + Just snat = someNatVal (fromIntegral n :: Integer) + +instance Arbitrary PermutationBraid where + arbitrary = do + perm <- arbitrary + return $ mkPermBraid perm + shrink (PermBraid x b) = [ PermBraid (braidPermutation s) s | s <- shrinkBraid b ] + +-------------------------------------------------------------------------------- +-- * test groups + +testgroup_Braid :: Test +testgroup_Braid = testGroup "Braid" + + [ testProperty "linking matrix is invariant of reduction" prop_link_reduce + , testProperty "linking matrix is invariant of perturbation" prop_link_perturb + + , testProperty "tau^2 = identity" prop_tau_square + , testProperty "tau commutes with braidPermutation" prop_permTau_1 + + , testProperty "braidPermutation . permutationBraid = identity" prop_permBraid_perm + , testProperty "permutation braid is indeed a permutation braid" prop_permBraid_valid + , testProperty "multiplication commutes with braidPermutation" prop_braidPerm_comp + + , testProperty "positive braids have positive links" prop_link_positive + , testProperty "definition of linking" prop_linking + + ] + +-------------------------------------------------------------------------------- + +testgroup_Braid_NF :: Test +testgroup_Braid_NF = testGroup "Braid/NF" + + [ testProperty "NF with naive inverse elimination == less naive inverse elimination" prop_braidnf_naive + , testProperty "NF with reduction == NF without reduction" prop_braidnf_reduce + + , testProperty "NF = NF of representative word of NF" prop_braidnf_reprs + , testProperty "NF = NF of perturbed word" prop_braidnf_perturb + + , testProperty "linking of word == linking of representative of NF" prop_braidnf_link + + , testProperty "NF of positive word is positive" prop_braidnf_pos + + , testProperty "Lemma 2.5" prop_lemma_2_5 + + , testProperty "permutationBraid and tau commutes, up to NF" prop_permTau_2 + ] + +-------------------------------------------------------------------------------- +-- * braid properties + +prop_link_reduce :: UnreducedBraid -> Bool +prop_link_reduce (Unreduced braid) = linkingMatrix braid == linkingMatrix braid' where + braid' = freeReduceBraidWord braid + +prop_link_perturb :: PerturbedBraid -> Bool +prop_link_perturb (Perturbed braid1 braid2) = linkingMatrix braid1 == linkingMatrix braid2 + +prop_tau_square :: ReducedBraid -> Bool +prop_tau_square (Reduced braid) = braidWord (tau (tau braid)) == braidWord braid + +prop_permTau_1 :: PermutationBraid -> Bool +prop_permTau_1 (PermBraid perm braid) = tauPerm perm == braidPermutation (tau braid) + +prop_permBraid_perm :: PermutationBraid -> Bool +prop_permBraid_perm (PermBraid perm braid) = (braidPermutation braid == perm) + +prop_permBraid_valid :: PermutationBraid -> Bool +prop_permBraid_valid (PermBraid perm braid) = isPermutationBraid braid + +prop_braidPerm_comp :: TwoBraids -> Bool +prop_braidPerm_comp (TwoBraids b1 b2) = (p == q) where + p = braidPermutation (compose b1 b2) + q = braidPermutation b1 `P.multiply` braidPermutation b2 + +prop_link_positive :: PositiveBraid -> Bool +prop_link_positive (PositiveB braid) = all (>=0) $ elems $ linkingMatrix braid + +prop_linking :: UnreducedBraid -> Bool +prop_linking (Unreduced braid) = (linkingMatrix braid == matrix) where + n = numberOfStrands braid + matrix = array ((1,1),(n,n)) [ ((i,j),strandLinking braid i j) | i<-[1..n], j<-[1..n] ] + +-------------------------------------------------------------------------------- + +prop_braidnf_naive :: UnreducedBraid -> Bool +prop_braidnf_naive (Unreduced braid) = (braidNormalFormNaive' braid == braidNormalForm' braid) + +prop_braidnf_reduce :: UnreducedBraid -> Bool +prop_braidnf_reduce (Unreduced braid) = (braidNormalForm' braid == braidNormalForm braid) + +prop_braidnf_reprs :: ReducedBraid -> Bool +prop_braidnf_reprs (Reduced braid) = (nf == nf') where + nf = braidNormalForm braid + nf' = braidNormalForm braid' + braid' = nfReprWord nf + +prop_braidnf_perturb :: PerturbedBraid -> Bool +prop_braidnf_perturb (Perturbed braid1 braid2) = (braidNormalForm braid1 == braidNormalForm braid2) + +prop_braidnf_link :: UnreducedBraid -> Bool +prop_braidnf_link (Unreduced braid) = (linkingMatrix braid == linkingMatrix braid') where + nf = braidNormalForm braid + braid' = nfReprWord nf + +prop_braidnf_pos :: PositiveBraid -> Bool +prop_braidnf_pos (PositiveB braid) = (_nfDeltaExp (braidNormalForm braid) >= 0) + +prop_lemma_2_5 :: Permutation -> Bool +prop_lemma_2_5 p = and [ check i | i<-[1..n-1] ] where + n = P.permutationSize p + w = _permutationBraid p + s = permWordStartingSet n w + check i = _isPermutationBraid n (i:w) == (not $ elem i s) + +prop_permTau_2 :: PermutationBraid -> Bool +prop_permTau_2 (PermBraid perm braid) = (nf1 == nf2) where + nf1 = braidNormalForm $ permutationBraid (tauPerm perm) + nf2 = braidNormalForm $ tau braid + +-------------------------------------------------------------------------------- + + diff --git a/test/Tests/Common.hs b/test/Tests/Common.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Common.hs @@ -0,0 +1,35 @@ + +-- | Helper routines for tests + +{-# LANGUAGE Rank2Types #-} +module Tests.Common where + +-------------------------------------------------------------------------------- + +import Test.QuickCheck +import Test.QuickCheck.Gen + +import System.Random + +-------------------------------------------------------------------------------- + +-- | Generates a random element. +choose_ :: Random a => Gen a +choose_ = MkGen (\r _ -> let (x,_) = random r in x) + +-- | Generates two random elements +choosePair_ :: Random a => Gen (a,a) +choosePair_ = do + x <- choose_ + y <- choose_ + return (x,y) + +-- | Generates a random element. +myMkGen :: (forall g. RandomGen g => g -> (a,g)) -> Gen a +myMkGen fun = MkGen (\r _ -> let (x,_) = fun r in x) + +-- | Generates a random element. +myMkSizedGen :: (forall g. RandomGen g => Int -> g -> (a,g)) -> Gen a +myMkSizedGen fun = MkGen (\r siz -> let (x,_) = fun siz r in x) + +-------------------------------------------------------------------------------- diff --git a/test/Tests/LatticePaths.hs b/test/Tests/LatticePaths.hs new file mode 100644 --- /dev/null +++ b/test/Tests/LatticePaths.hs @@ -0,0 +1,111 @@ + +-- | Tests for lattice paths +-- + +{-# LANGUAGE CPP, ScopedTypeVariables, GeneralizedNewtypeDeriving, FlexibleContexts #-} +module Tests.LatticePaths where + +-------------------------------------------------------------------------------- + +import Math.Combinat.LatticePaths + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import System.Random + +import Control.Monad + +import Data.List + +import Math.Combinat.Classes +import Math.Combinat.Helper +import Math.Combinat.Sign +import Math.Combinat.Numbers ( factorial , binomial ) + +-------------------------------------------------------------------------------- +-- * instances + +-- | Half-length of a Dyck path +newtype Half = Half Int deriving (Eq,Show) + +-- | First number is (usually) less or equal than the second +data HalfPair = HalfPair Int Int deriving (Eq,Show) + +maxHalfSize :: Int +maxHalfSize = 11 -- number of paths grow exponentially + +instance Arbitrary Half where + arbitrary = liftM Half $ choose (0,maxHalfSize) + +instance Arbitrary HalfPair where + arbitrary = do + n <- choose (0,maxHalfSize) + k <- choose (0,n+1) + return (HalfPair k n) + +fi :: Int -> Integer +fi = fromIntegral + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_LatticePaths :: Test +testgroup_LatticePaths = testGroup "Lattice paths" + + [ testProperty "dyck paths are in reverse lexicographic order" prop_revlex + , testProperty "naive Dyck path algorithm = less naive algorithm" prop_dyck_naive + , testProperty "counting Dyck paths" prop_count + , testProperty "counting Lattice paths" prop_count_lattice + + , testProperty "bounded Dyck paths, def, v1" prop_bounded_1 + , testProperty "bounded Dyck paths, def, v2" prop_bounded_2 + , testProperty "bounded Dyck paths w/ high bound = all dyck paths" prop_not_bounded + + , testProperty "zero-touching Dyck paths" prop_touching + , testProperty "Dyck paths w/ k peaks" prop_peaking + + ] + +-------------------------------------------------------------------------------- +-- * test properties + +prop_revlex :: Bool +prop_revlex = and [ sort (dyckPaths m) == reverse (dyckPaths m) | m <- [0..maxHalfSize] ] + +prop_dyck_naive :: Bool +prop_dyck_naive = and [ sort (dyckPathsNaive m) == sort (dyckPaths m) | m <- [0..maxHalfSize] ] + +prop_count :: Bool +prop_count = and [ fi (length (dyckPaths m)) == countDyckPaths m | m <- [0..maxHalfSize] ] + +prop_count_lattice :: HalfPair -> Bool +prop_count_lattice (HalfPair y x) = fi (length (latticePaths (x,y))) == countLatticePaths (x,y) + +prop_bounded_1 :: HalfPair -> Bool +prop_bounded_1 (HalfPair h m) = (one == two) where + one = sort (boundedDyckPaths h m ) + two = sort [ p | p <- dyckPaths m , pathHeight p <= h ] + +prop_bounded_2 :: Half -> Half -> Bool +prop_bounded_2 (Half h) (Half m) = (one == two) where + one = sort (boundedDyckPaths h m ) + two = sort [ p | p <- dyckPaths m , pathHeight p <= h ] + +prop_not_bounded :: Bool +prop_not_bounded = and [ sort (boundedDyckPaths m m) == sort (dyckPaths m) | m <- [0..maxHalfSize] ] + +prop_touching :: HalfPair -> Bool +prop_touching (HalfPair k m) = (one == two && fi (length one) == cnt) where + one = sort (touchingDyckPaths k m) + two = sort [ p | p <- dyckPaths m , pathNumberOfZeroTouches p == k ] + cnt = countTouchingDyckPaths k m + +prop_peaking :: HalfPair -> Bool +prop_peaking (HalfPair k m) = (one == two && fi (length one) == cnt) where + one = sort (peakingDyckPaths k m) + two = sort [ p | p <- dyckPaths m , pathNumberOfPeaks p == k ] + cnt = countPeakingDyckPaths k m + +-------------------------------------------------------------------------------- + diff --git a/test/Tests/Partitions/Integer.hs b/test/Tests/Partitions/Integer.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Partitions/Integer.hs @@ -0,0 +1,107 @@ + +-- | Tests for integer partitions. + +{-# LANGUAGE CPP, BangPatterns #-} +module Tests.Partitions.Integer where + +-------------------------------------------------------------------------------- + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck + +import Tests.Common + +import Math.Combinat.Partitions.Integer + +import Data.List +import Control.Monad + +-- import Data.Map (Map) +-- import qualified Data.Map as Map + +import Math.Combinat.Classes +import Math.Combinat.Numbers ( factorial , binomial , multinomial ) +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * Types and instances + +newtype PartitionWeight = PartitionWeight Int deriving (Eq,Show) +data PartitionWeightPair = PartitionWeightPair Int Int deriving (Eq,Show) +data PartitionIntPair = PartitionIntPair Partition Int deriving (Eq,Show) + +maxPartitionSize :: Int +maxPartitionSize = 44 + +instance Arbitrary Partition where + arbitrary = do + n <- choose (0,maxPartitionSize) + myMkGen (randomPartition n) + +instance Arbitrary PartitionWeight where + arbitrary = liftM PartitionWeight $ choose (0,maxPartitionSize) + +instance Arbitrary PartitionWeightPair where + arbitrary = do + n <- choose (0,maxPartitionSize) + k <- choose (0,n+2) + return (PartitionWeightPair n k) + +instance Arbitrary PartitionIntPair where + arbitrary = do + part <- arbitrary + k <- choose (0,partitionWeight part + 2) + return (PartitionIntPair part k) + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_IntegerPartitions :: Test +testgroup_IntegerPartitions = testGroup "Integer Partitions" + + [ testProperty "partitions in a box" prop_partitions_in_bigbox + , testProperty "partitions with k parts" prop_kparts + , testProperty "odd partitions" prop_odd_partitions + , testProperty "partitions with distinct parts" prop_distinct_partitions + , testProperty "subpartitions" prop_subparts + , testProperty "dual^2 is identity" prop_dual_dual + , testProperty "dominated partitions" prop_dominated_list + , testProperty "dominating partitions" prop_dominating_list + , testProperty "counting partitions" prop_countParts + ] + +-------------------------------------------------------------------------------- +-- * properties + +prop_partitions_in_bigbox :: PartitionWeight -> Bool +prop_partitions_in_bigbox (PartitionWeight n) = (partitions n == partitions' (n,n) n) + +prop_kparts :: PartitionWeightPair -> Bool +prop_kparts (PartitionWeightPair n k) = (partitionsWithKParts k n == [ mu | mu <- partitions n, numberOfParts mu == k ]) + +prop_odd_partitions :: PartitionWeight -> Bool +prop_odd_partitions (PartitionWeight n) = + (partitionsWithOddParts n == [ mu | mu <- partitions n, and (map odd (fromPartition mu)) ]) + +prop_distinct_partitions :: PartitionWeight -> Bool +prop_distinct_partitions (PartitionWeight n) = + (partitionsWithDistinctParts n == [ mu | mu <- partitions n, let xs = fromPartition mu, xs == nub xs ]) + +prop_subparts :: PartitionIntPair -> Bool +prop_subparts (PartitionIntPair lam d) = (subPartitions d lam) == sort [ p | p <- partitions d, isSubPartitionOf p lam ] + +prop_dual_dual :: Partition -> Bool +prop_dual_dual lam = (lam == dualPartition (dualPartition lam)) + +prop_dominated_list :: Partition -> Bool +prop_dominated_list lam = (dominatedPartitions lam == [ mu | mu <- partitions (weight lam), lam `dominates` mu ]) + +prop_dominating_list :: Partition -> Bool +prop_dominating_list mu = (dominatingPartitions mu == [ lam | lam <- partitions (weight mu ), lam `dominates` mu ]) + +prop_countParts :: Bool +prop_countParts = (take 50 partitionCountList == take 50 partitionCountListNaive) + +-------------------------------------------------------------------------------- + diff --git a/test/Tests/Partitions/Skew.hs b/test/Tests/Partitions/Skew.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Partitions/Skew.hs @@ -0,0 +1,85 @@ + +-- | Tests for skew partitions. +-- + +{-# LANGUAGE CPP, BangPatterns #-} +module Tests.Partitions.Skew where + +-------------------------------------------------------------------------------- + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck + +import Tests.Common +import Tests.Partitions.Integer () -- Arbitrary instances + +import Math.Combinat.Partitions.Integer +import Math.Combinat.Partitions.Skew + +import Data.List + +import Math.Combinat.Classes + +-------------------------------------------------------------------------------- +-- * instances + +instance Arbitrary SkewPartition where + arbitrary = do + p <- arbitrary + let n = partitionWeight p + d <- choose (0,n) + let qs = subPartitions d p + ln = length qs + k <- choose (0,ln-1) + let q = qs !! k + return $ mkSkewPartition (p,q) + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_SkewPartitions :: Test +testgroup_SkewPartitions = testGroup "Skew Partitions" + + [ testProperty "dual^2 = identity" prop_dual_dual + , testProperty "dual vs. inner/outer dual" prop_dual_from + , testProperty "to . from = identity" prop_from_to + , testProperty "from . to = identity" prop_to_from + , testProperty "from . to . from = from" prop_from_to_from + , testProperty "weight vs. inner/outer weight" prop_weight + ] + +-------------------------------------------------------------------------------- +-- * properties + +prop_dual_dual :: SkewPartition -> Bool +prop_dual_dual sp = (dualSkewPartition (dualSkewPartition sp) == sp) + +prop_dual_from :: SkewPartition -> Bool +prop_dual_from sp = (p == dual p' && q == dual q') where + (p,q) = fromSkewPartition sp + sp' = dualSkewPartition sp + (p',q') = fromSkewPartition sp' + +prop_from_to :: SkewPartition -> Bool +prop_from_to sp = (mkSkewPartition (fromSkewPartition sp) == sp) + +prop_to_from :: (Partition,Partition) -> Bool +prop_to_from (p,q) = + case mb of + Nothing -> True + Just sp -> fromSkewPartition sp == (p,q) + where + mb = safeSkewPartition (p,q) + +prop_from_to_from :: SkewPartition -> Bool +prop_from_to_from sp = (pq == pq') where + pq = fromSkewPartition sp + sp' = mkSkewPartition pq + pq' = fromSkewPartition sp' + +prop_weight :: SkewPartition -> Bool +prop_weight sp = (skewPartitionWeight sp == weight p - weight q) where + (p,q) = fromSkewPartition sp + +-------------------------------------------------------------------------------- diff --git a/test/Tests/Permutations.hs b/test/Tests/Permutations.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Permutations.hs @@ -0,0 +1,219 @@ + +-- | Tests for permutations. +-- + +{-# LANGUAGE CPP, ScopedTypeVariables, GeneralizedNewtypeDeriving, FlexibleContexts #-} +module Tests.Permutations where + +-------------------------------------------------------------------------------- + +import Math.Combinat.Permutations + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import System.Random + +import Control.Monad +import Control.Monad.ST + +import Data.List hiding (permutations) + +import Data.Array (Array) +import Data.Array.ST +import Data.Array.Unboxed +import Data.Array.IArray +import Data.Array.MArray +import Data.Array.Unsafe + +import Math.Combinat.ASCII +import Math.Combinat.Classes +import Math.Combinat.Helper +import Math.Combinat.Sign +import Math.Combinat.Numbers (factorial,binomial) + +-------------------------------------------------------------------------------- +-- * generating permutations (random & arbitrary instances, spec types etc) + +minPermSize = 1 +maxPermSize = 123 + +newtype Elem = Elem Int deriving Eq +newtype Nat = Nat { fromNat :: Int } deriving (Eq,Ord,Show,Num,Random) + +naturalSet :: Permutation -> Array Int Elem +naturalSet perm = listArray (1,n) [ Elem i | i<-[1..n] ] where + n = permutationSize perm + +permInternalSet :: Permutation -> Array Int Elem +permInternalSet perm@(Permutation arr) = listArray (1,n) [ Elem (arr!i) | i<-[1..n] ] where + n = permutationSize perm + +sameSize :: Permutation -> Permutation -> Bool +sameSize perm1 perm2 = ( permutationSize perm1 == permutationSize perm2) + +newtype CyclicPermutation = Cyclic { fromCyclic :: Permutation } deriving Show + +data SameSize = SameSize Permutation Permutation deriving Show + +instance Random Permutation where + random g = randomPermutation size g1 where + (size,g1) = randomR (minPermSize,maxPermSize) g + randomR _ = random + +instance Random CyclicPermutation where + random g = (Cyclic cycl,g2) where + (size,g1) = randomR (minPermSize,maxPermSize) g + (cycl,g2) = randomCyclicPermutation size g1 + randomR _ = random + +instance Random DisjointCycles where + random g = (disjcyc,g2) where + (size,g1) = randomR (minPermSize,maxPermSize) g + (perm,g2) = randomPermutation size g1 + disjcyc = permutationToDisjointCycles perm + randomR _ = random + +instance Random SameSize where + random g = (SameSize prm1 prm2, g3) where + (size,g1) = randomR (minPermSize,maxPermSize) g + (prm1,g2) = randomPermutation size g1 + (prm2,g3) = randomPermutation size g2 + randomR _ = random + +instance Arbitrary Nat where + arbitrary = choose (Nat 0 , Nat 50) + +instance Arbitrary Permutation where arbitrary = choose undefined +instance Arbitrary CyclicPermutation where arbitrary = choose undefined +instance Arbitrary DisjointCycles where arbitrary = choose undefined +instance Arbitrary SameSize where arbitrary = choose undefined + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_Permutations :: Test +testgroup_Permutations = testGroup "Permutations" + + [ testProperty "disjoint cycles /1" prop_disjcyc_1 + , testProperty "disjoint cycles /2" prop_disjcyc_2 + + , testProperty "disjoint cycles compatibility" prop_disjcyc_Mathematica + + , testProperty "random cyclic permutation is indeed cyclic" prop_randCyclic + , testProperty "inverse^2 is identity" prop_inverse + + , testProperty "permutation action is group action" prop_mulPerm + , testProperty "left permutation action is left group action" prop_mulPermLeft + , testProperty "right permutation action is right group action" prop_mulPermRight + + , testProperty "permutation action convetion" prop_perm + , testProperty "left permutation action convention" prop_permLeft + , testProperty "right permutation action convention" prop_permRight + , testProperty "left/right permutation action convention" prop_permLeftRight + + , testProperty "cycle left" prop_cycleLeft + , testProperty "cycle right" prop_cycleRight + + , testProperty "sign of permutation is multiplicative" prop_mulSign + , testProperty "inverse is compatible with multiplication" prop_invMul + + , testProperty "parity of cyclic permutaiton" prop_cyclSign + , testProperty "random permutation is valid" prop_permIsPerm + , testProperty "definition of parity" prop_isEven + + , testProperty "bubbleSort works" prop_bubbleSort + , testProperty "bubbleSort2 works" prop_bubbleSort2 + , testProperty "number of inversions = steps in bubble sort" prop_bubble_inversions + , testProperty "number of inversions = actual number of inversions" prop_number_inversions + , testProperty "number of inversions is the same for the inverse permutation" prop_ninversions_inverse + , testProperty "merge sort algorithm = naive inversion count" prop_merge_inversions + + ] + +-------------------------------------------------------------------------------- +-- * test properties + +prop_disjcyc_1 perm = ( perm == disjointCyclesToPermutation n (permutationToDisjointCycles perm) ) + where n = permutationSize perm + +prop_disjcyc_2 k dcyc = ( dcyc == permutationToDisjointCycles (disjointCyclesToPermutation n dcyc) ) + where + n = fromNat k + m + m = case fromDisjointCycles dcyc of + [] -> 1 + xxs -> maximum (concat xxs) + +-- PermutationCycles[ { 12, 15, 5, 6, 2, 7, 17, 9, 20, 3, 11, 18, 22, 21, 8, 10, 4, 19, 14, 16, 23, 1, 13 } ] +-- Cycles [ {{1, 12, 18, 19, 14, 21, 23, 13, 22}, {2, 15, 8, 9, 20, 16, 10, 3, 5}, {4, 6, 7, 17}} ] +prop_disjcyc_Mathematica = (permutationToDisjointCycles perm == disjcyc) + && (disjointCyclesToPermutation n disjcyc == perm) + where + n = permutationSize perm + perm = toPermutation [ 12, 15, 5, 6, 2, 7, 17, 9, 20, 3, 11, 18, 22, 21, 8, 10, 4, 19, 14, 16, 23, 1, 13 ] + disjcyc = DisjointCycles [ [1, 12, 18, 19, 14, 21, 23, 13, 22], [2, 15, 8, 9, 20, 16, 10, 3, 5], [4, 6, 7, 17] ] + +xperm = toPermutation [ 12, 15, 5, 6, 2, 7, 17, 9, 20, 3, 11, 18, 22, 21, 8, 10, 4, 19, 14, 16, 23, 1, 13 ] +xdisjcyc = DisjointCycles [ [1, 12, 18, 19, 14, 21, 23, 13, 22], [2, 15, 8, 9, 20, 16, 10, 3, 5], [4, 6, 7, 17] ] + +prop_randCyclic cycl = ( isCyclicPermutation (fromCyclic cycl) ) + +prop_inverse perm = ( perm == inverse (inverse perm) ) + +prop_mulPerm (SameSize perm1 perm2) = + ( permute perm2 (permute perm1 set) == permute (perm1 `multiply` perm2) set ) + where + set = naturalSet perm1 + +prop_mulPermRight (SameSize perm1 perm2) = + ( permuteRight perm2 (permuteRight perm1 set) == permuteRight (perm1 `multiply` perm2) set ) + where + set = naturalSet perm1 + +prop_mulPermLeft (SameSize perm1 perm2) = + ( permuteLeft perm2 (permuteLeft perm1 set) == permuteLeft (perm2 `multiply` perm1) set ) + where + set = naturalSet perm1 + +prop_perm perm = permute perm (naturalSet perm) == permInternalSet perm +prop_permLeft perm = permuteLeft perm (permInternalSet perm) == naturalSet perm +prop_permRight perm = permuteRight perm (naturalSet perm) == permInternalSet perm +prop_permLeftRight perm = permuteLeft (inverse perm) (naturalSet perm) == permuteRight (perm) (naturalSet perm) + +prop_cycleLeft = permuteList (cycleLeft 5) "abcde" == "bcdea" +prop_cycleRight = permuteList (cycleRight 5) "abcde" == "eabcd" + +prop_mulSign (SameSize perm1 perm2) = + ( sgn perm1 * sgn perm2 == sgn (perm1 `multiply` perm2) ) + where + sgn = signValue . signOfPermutation :: Permutation -> Int + +prop_invMul (SameSize perm1 perm2) = + ( inverse perm2 `multiply` inverse perm1 == inverse (perm1 `multiply` perm2) ) + +prop_cyclSign cycl = ( isEvenPermutation perm == odd n ) where + perm = fromCyclic cycl + n = permutationSize perm + +prop_permIsPerm perm = ( isPermutation (fromPermutation perm) ) + +prop_isEven perm = ( isEvenPermutation perm == isEvenAlternative perm ) where + isEvenAlternative p = + even $ sum $ map (\x->x-1) $ map length $ fromDisjointCycles $ permutationToDisjointCycles p + +prop_bubbleSort perm = multiplyMany' n (map (adjacentTransposition n) $ bubbleSort perm) == perm where + n = permutationSize perm + +prop_bubbleSort2 perm = multiplyMany' n (map (transposition n) $ bubbleSort2 perm) == perm where + n = permutationSize perm + +prop_bubble_inversions perm = length (bubbleSort perm) == numberOfInversions perm + +prop_number_inversions perm = length (inversions perm) == numberOfInversions perm + +prop_ninversions_inverse perm = numberOfInversions perm == numberOfInversions (inverse perm) + +prop_merge_inversions perm = (numberOfInversionsMerge perm == numberOfInversionsNaive perm) + +-------------------------------------------------------------------------------- + diff --git a/test/Tests/Series.hs b/test/Tests/Series.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Series.hs @@ -0,0 +1,303 @@ + +-- | Tests for power series +-- + +{-# LANGUAGE CPP, GeneralizedNewtypeDeriving #-} +module Tests.Series where + +-------------------------------------------------------------------------------- + +import Math.Combinat.Numbers.Series + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import System.Random + +import Data.List + +import Math.Combinat.Sign +import Math.Combinat.Numbers +import Math.Combinat.Partitions.Integer +import Math.Combinat.Helper + +-------------------------------------------------------------------------------- +-- * code used only for tests + +-- | Expansion of @1 / (1-x^k)@. Included for completeness only; +-- it equals to @coinSeries [k]@, and for example +-- for @k=4@ it is simply +-- +-- > [1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0...] +-- +pseries1 :: Int -> [Integer] +pseries1 k1 = convolveWithPSeries1 k1 unitSeries + +-- | The expansion of @1 / (1-x^k_1-x^k_2)@ +pseries2 :: Int -> Int -> [Integer] +pseries2 k1 k2 = convolveWithPSeries2 k1 k2 unitSeries + +-- | The expansion of @1 / (1-x^k_1-x^k_2-x^k_3)@ +pseries3 :: Int -> Int -> Int -> [Integer] +pseries3 k1 k2 k3 = convolveWithPSeries3 k1 k2 k3 unitSeries + +-------------------------------------------------------------------------------- + +-- | Convolve with (the expansion of) @1 / (1-x^k1)@ +convolveWithPSeries1 :: Int -> [Integer] -> [Integer] +convolveWithPSeries1 k1 series1 = xs where + series = series1 ++ repeat 0 + xs = zipWith (+) series ( replicate k1 0 ++ xs ) + +-- | Convolve with (the expansion of) @1 / (1-x^k1-x^k2)@ +convolveWithPSeries2 :: Int -> Int -> [Integer] -> [Integer] +convolveWithPSeries2 k1 k2 series1 = xs where + series = series1 ++ repeat 0 + xs = zipWith3 (\x y z -> x + y + z) + series + ( replicate k1 0 ++ xs ) + ( replicate k2 0 ++ xs ) + +-- | Convolve with (the expansion of) @1 / (1-x^k_1-x^k_2-x^k_3)@ +convolveWithPSeries3 :: Int -> Int -> Int -> [Integer] -> [Integer] +convolveWithPSeries3 k1 k2 k3 series1 = xs where + series = series1 ++ repeat 0 + xs = zipWith4 (\x y z w -> x + y + z + w) + series + ( replicate k1 0 ++ xs ) + ( replicate k2 0 ++ xs ) + ( replicate k3 0 ++ xs ) + +-------------------------------------------------------------------------------- + +-- | @1 / (1 - a*x^k)@. +-- For example, for @a=3@ and @k=2@ it is just +-- +-- > [1,0,3,0,9,0,27,0,81,0,243,0,729,0,2187,0,6561,0,19683,0...] +-- +pseries1' :: Num a => (a,Int) -> [a] +pseries1' ak1 = convolveWithPSeries1' ak1 unitSeries + +-- | @1 / (1 - a_1*x^k_1 - a_2*x^k_2)@ +pseries2' :: Num a => (a,Int) -> (a,Int) -> [a] +pseries2' ak1 ak2 = convolveWithPSeries2' ak1 ak2 unitSeries + +-- | @1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3)@ +pseries3' :: Num a => (a,Int) -> (a,Int) -> (a,Int) -> [a] +pseries3' ak1 ak2 ak3 = convolveWithPSeries3' ak1 ak2 ak3 unitSeries + +-------------------------------------------------------------------------------- + +-- | Convolve with @1 / (1 - a*x^k)@. +convolveWithPSeries1' :: Num a => (a,Int) -> [a] -> [a] +convolveWithPSeries1' (a1,k1) series1 = xs where + series = series1 ++ repeat 0 + xs = zipWith (+) + series + ( replicate k1 0 ++ map (*a1) xs ) + +-- | Convolve with @1 / (1 - a_1*x^k_1 - a_2*x^k_2)@ +convolveWithPSeries2' :: Num a => (a,Int) -> (a,Int) -> [a] -> [a] +convolveWithPSeries2' (a1,k1) (a2,k2) series1 = xs where + series = series1 ++ repeat 0 + xs = zipWith3 (\x y z -> x + y + z) + series + ( replicate k1 0 ++ map (*a1) xs ) + ( replicate k2 0 ++ map (*a2) xs ) + +-- | Convolve with @1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3)@ +convolveWithPSeries3' :: Num a => (a,Int) -> (a,Int) -> (a,Int) -> [a] -> [a] +convolveWithPSeries3' (a1,k1) (a2,k2) (a3,k3) series1 = xs where + series = series1 ++ repeat 0 + xs = zipWith4 (\x y z w -> x + y + z + w) + series + ( replicate k1 0 ++ map (*a1) xs ) + ( replicate k2 0 ++ map (*a2) xs ) + ( replicate k3 0 ++ map (*a3) xs ) + +-------------------------------------------------------------------------------- +-- * Types and instances + +{- +swap :: (a,b) -> (b,a) +swap (x,y) = (y,x) +-} + +-- compare the first 500 elements of the infinite lists +(=!=) :: (Eq a, Num a) => [a] -> [a] -> Bool +(=!=) xs1 ys1 = (take m xs == take m ys) where + m = 500 + xs = xs1 ++ repeat 0 + ys = ys1 ++ repeat 0 + +infix 4 =!= + +newtype Nat = Nat { fromNat :: Int } deriving (Eq,Ord,Show,Num,Random) +newtype Ser = Ser { fromSer :: [Integer] } deriving (Eq,Ord,Show) +newtype Exp = Exp { fromExp :: Int } deriving (Eq,Ord,Show,Num,Random) +newtype Exps = Exps { fromExps :: [Int] } deriving (Eq,Ord,Show) +newtype CoeffExp = CoeffExp { fromCoeffExp :: (Integer,Int) } deriving (Eq,Ord,Show) +newtype CoeffExps = CoeffExps { fromCoeffExps :: [(Integer,Int)] } deriving (Eq,Ord,Show) + +minSerSize = 0 :: Int +maxSerSize = 1000 :: Int + +minSerValue = -10000 :: Int +maxSerValue = 10000 :: Int + +rndList :: (RandomGen g, Random a) => Int -> (a, a) -> g -> ([a], g) +rndList n minmax g = swap $ mapAccumL f g [1..n] where + f g _ = swap $ randomR minmax g + +instance Arbitrary Nat where + arbitrary = choose (Nat 0 , Nat 750) + +instance Arbitrary Exp where + arbitrary = choose (Exp 1 , Exp 32) + +instance Arbitrary CoeffExp where + arbitrary = do + coeff <- choose (minSerValue, maxSerValue) :: Gen Int + exp <- arbitrary :: Gen Exp + return $ CoeffExp (fromIntegral coeff, fromExp exp) + +instance Random Ser where + random g = (Ser $ map fi list, g2) where + (size,g1) = randomR (minSerSize,maxSerSize) g + (list,g2) = rndList size (minSerValue,maxSerValue) g1 + fi :: Int -> Integer + fi = fromIntegral + randomR _ = random + +instance Random Exps where + random g = (Exps list, g2) where + (size,g1) = randomR (0,10) g + (list,g2) = rndList size (1,32) g1 + randomR _ = random + +instance Random CoeffExps where + random g = (CoeffExps (zip (map fromIntegral list2) list1), g3) where + (size,g1) = randomR (0,10) g + (list1,g2) = rndList size (1,32) g1 + (list2,g3) = rndList size (minSerValue,maxSerValue) g2 + randomR _ = random + +instance Arbitrary Ser where + arbitrary = choose undefined + +instance Arbitrary Exps where + arbitrary = choose undefined + +instance Arbitrary CoeffExps where + arbitrary = choose undefined + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_PowerSeries :: Test +testgroup_PowerSeries = testGroup "Power series" + [ + testProperty "convPSeries1 vs generic" prop_conv1_vs_gen + , testProperty "convPSeries2 vs generic" prop_conv2_vs_gen + , testProperty "convPSeries3 vs generic" prop_conv3_vs_gen + , testProperty "convPSeries1' vs generic" prop_conv1_vs_gen' + , testProperty "convPSeries2' vs generic" prop_conv2_vs_gen' + , testProperty "convPSeries3' vs generic" prop_conv3_vs_gen' + , testProperty "convolve_pseries" prop_convolve_pseries + , testProperty "convolve_pseries'" prop_convolve_pseries' + , testProperty "coinSeries vs pseries" prop_coin_vs_pseries + , testProperty "coinSeries vs pseries'" prop_coin_vs_pseries' + + -- these are very slow, because random is slow + , testProperty "leftIdentity" prop_leftIdentity + , testProperty "rightIdentity" prop_rightIdentity + , testProperty "commutativity" prop_commutativity + , testProperty "associativity" prop_associativity + ] + +-------------------------------------------------------------------------------- +-- * properties + +prop_leftIdentity ser = ( xs =!= unitSeries `convolve` xs ) where + xs = fromSer ser + +prop_rightIdentity ser = ( unitSeries `convolve` xs =!= xs ) where + xs = fromSer ser + +prop_commutativity ser1 ser2 = ( xs `convolve` ys =!= ys `convolve` xs ) where + xs = fromSer ser1 + ys = fromSer ser2 + +prop_associativity ser1 ser2 ser3 = ( one =!= two ) where + one = (xs `convolve` ys) `convolve` zs + two = xs `convolve` (ys `convolve` zs) + xs = fromSer ser1 + ys = fromSer ser2 + zs = fromSer ser3 + +prop_conv1_vs_gen exp1 ser = ( one =!= two ) where + one = convolveWithPSeries1 k1 xs + two = convolveWithPSeries [k1] xs + k1 = fromExp exp1 + xs = fromSer ser + +prop_conv2_vs_gen exp1 exp2 ser = (one =!= two) where + one = convolveWithPSeries2 k1 k2 xs + two = convolveWithPSeries [k2,k1] xs + k1 = fromExp exp1 + k2 = fromExp exp2 + xs = fromSer ser + +prop_conv3_vs_gen exp1 exp2 exp3 ser = (one =!= two) where + one = convolveWithPSeries3 k1 k2 k3 xs + two = convolveWithPSeries [k2,k3,k1] xs + k1 = fromExp exp1 + k2 = fromExp exp2 + k3 = fromExp exp3 + xs = fromSer ser + +prop_conv1_vs_gen' exp1 ser = ( one =!= two ) where + one = convolveWithPSeries1' ak1 xs + two = convolveWithPSeries' [ak1] xs + ak1 = fromCoeffExp exp1 + xs = fromSer ser + +prop_conv2_vs_gen' exp1 exp2 ser = (one =!= two) where + one = convolveWithPSeries2' ak1 ak2 xs + two = convolveWithPSeries' [ak2,ak1] xs + ak1 = fromCoeffExp exp1 + ak2 = fromCoeffExp exp2 + xs = fromSer ser + +prop_conv3_vs_gen' exp1 exp2 exp3 ser = (one =!= two) where + one = convolveWithPSeries3' ak1 ak2 ak3 xs + two = convolveWithPSeries' [ak2,ak3,ak1] xs + ak1 = fromCoeffExp exp1 + ak2 = fromCoeffExp exp2 + ak3 = fromCoeffExp exp3 + xs = fromSer ser + +prop_convolve_pseries exps1 ser = (one =!= two) where + one = convolveWithPSeries ks1 xs + two = xs `convolve` pseries ks1 + ks1 = fromExps exps1 + xs = fromSer ser + +prop_convolve_pseries' cexps1 ser = (one =!= two) where + one = convolveWithPSeries' aks1 xs + two = xs `convolve` pseries' aks1 + aks1 = fromCoeffExps cexps1 + xs = fromSer ser + +prop_coin_vs_pseries exps1 = (one =!= two) where + one = coinSeries ks1 + two = convolveMany (map pseries1 ks1) + ks1 = fromExps exps1 + +prop_coin_vs_pseries' cexps1 = (one =!= two) where + one = coinSeries' aks1 + two = convolveMany (map pseries1' aks1) + aks1 = fromCoeffExps cexps1 + +-------------------------------------------------------------------------------- + diff --git a/test/Tests/SkewTableaux.hs b/test/Tests/SkewTableaux.hs new file mode 100644 --- /dev/null +++ b/test/Tests/SkewTableaux.hs @@ -0,0 +1,103 @@ + +-- | Tests for skew tableaux + +{-# LANGUAGE FlexibleInstances #-} +module Tests.SkewTableaux where + +-------------------------------------------------------------------------------- + +import Control.Monad + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import Test.QuickCheck.Gen + +import Tests.Partitions.Integer () +import Tests.Partitions.Skew () -- arbitrary instances + +import Math.Combinat.Tableaux +import Math.Combinat.Tableaux.Skew +import Math.Combinat.Partitions.Integer +import Math.Combinat.Partitions.Skew + +-------------------------------------------------------------------------------- +-- * code + +numberOfNonEmptyRows :: SkewPartition -> Int +numberOfNonEmptyRows (SkewPartition xys) = length [ True | (x,y) <- xys, y/=0 ] + +-- | Breaks a skew partition into disjoint parts +disjointParts :: SkewPartition -> [SkewPartition] +disjointParts (SkewPartition xys) = map normalizeSkewPartition list where + + list = map SkewPartition $ filter (not . isEmpty) $ break xys + + isEmpty :: [(Int,Int)] -> Bool + isEmpty xys = and [ y==0 | (x,y) <- xys ] + + break :: [(Int,Int)] -> [[(Int,Int)]] + break [] = [[]] + break [xy] = [[xy]] + break ( xy@(x,y) : rest@((x',y'):_) ) = if x >= x'+y' + then [xy] : break rest + else let ( xys : rest' ) = break rest + in ( (xy:xys) : rest' ) + + + + +-------------------------------------------------------------------------------- +-- * instances + +instance Arbitrary (SkewTableau Int) where + arbitrary = do + shape <- arbitrary + let w = skewPartitionWeight shape + content <- replicateM w $ choose (1,1000) + return $ fillSkewPartitionWithRowWord shape content + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_SkewTableaux :: Test +testgroup_SkewTableaux = testGroup "Skew tableaux" + [ testProperty "dual^2 = identity" prop_skew_dual_dual + , testProperty "fill . rowWord = identity" prop_rowWord + , testProperty "fill . columnWord = identity" prop_columnWord + , testProperty "fill respectes the shape" prop_fill_shape + , testProperty "semistandard skew tableaux are indeed semistandard" prop_semistandard + ] + +-------------------------------------------------------------------------------- +-- * properties + +prop_skew_dual_dual :: SkewTableau Int -> Bool +prop_skew_dual_dual st = (dualSkewTableau (dualSkewTableau st) == st) + +prop_rowWord :: SkewTableau Int -> Bool +prop_rowWord st = (fillSkewPartitionWithRowWord shape content == st) where + shape = skewTableauShape st + content = skewTableauRowWord st + +prop_columnWord :: SkewTableau Int -> Bool +prop_columnWord st = (fillSkewPartitionWithColumnWord shape content == st) where + shape = skewTableauShape st + content = skewTableauColumnWord st + +prop_fill_shape :: SkewPartition -> Bool +prop_fill_shape shape = (shape == shape') where + tableau = fillSkewPartitionWithColumnWord shape [1..] + shape' = skewTableauShape tableau + +prop_semistandard :: SkewPartition -> Bool +prop_semistandard shape = and + [ isSemiStandardSkewTableau st + | n <- [kk..nn] + , st <- take 500 (semiStandardSkewTableaux n shape) -- we only take the first 500 because impossibly slow otherwise + ] + where + nn = min (kk + 10) (skewPartitionWeight shape) + kk = maximum $ 0 : (map numberOfNonEmptyRows $ disjointParts shape) + +-------------------------------------------------------------------------------- diff --git a/test/Tests/Thompson.hs b/test/Tests/Thompson.hs new file mode 100644 --- /dev/null +++ b/test/Tests/Thompson.hs @@ -0,0 +1,134 @@ + +-- | Tests for Thompson's group F +-- + +{-# LANGUAGE CPP, GeneralizedNewtypeDeriving, FlexibleInstances, TypeSynonymInstances #-} +module Tests.Thompson where + +-------------------------------------------------------------------------------- + +import Prelude hiding ( (**) ) +import Control.Monad +import Data.List + +import Math.Combinat.Groups.Thompson.F +import qualified Math.Combinat.Trees.Binary as B + +import Tests.Common + +import Test.Framework +import Test.Framework.Providers.QuickCheck2 +import Test.QuickCheck +import System.Random + +import Math.Combinat.Helper + + +-------------------------------------------------------------------------------- +-- * code + +(**) :: TDiag -> TDiag -> TDiag +(**) x y = x `compose` y + +(//) :: TDiag -> TDiag -> TDiag +(//) x y = x `compose` (inverse y) + +growth_n_sphere = [1,4,12,36,108,314,906,2576,7280,20352] :: [Int] +growth_pos_n_sphere = [1,2, 4, 9, 20, 45,101, 227, 510, 1146] :: [Int] + +-------------------------------------------------------------------------------- +-- * instances + +-- | A pair of trees with the same size +data TPair = TPair !T !T deriving (Eq,Show) + +newtype Unreduced = Unreduced TDiag deriving (Eq,Show) + +instance Arbitrary T where + arbitrary = liftM fromBinTree $ myMkSizedGen B.randomBinaryTree + +instance Arbitrary TPair where + arbitrary = myMkSizedGen $ \siz -> runRand $ do + t1 <- rand (B.randomBinaryTree siz) + t2 <- rand (B.randomBinaryTree siz) + return $ TPair (fromBinTree t1) (fromBinTree t2) + +instance Arbitrary TDiag where + arbitrary = do + TPair t1 t2 <- arbitrary + return $ mkTDiag t1 t2 + +instance Arbitrary Unreduced where + arbitrary = do + TPair t1 t2 <- arbitrary + return $ Unreduced $ mkTDiagDontReduce t1 t2 + +-------------------------------------------------------------------------------- +-- * test group + +testgroup_ThompsonF :: Test +testgroup_ThompsonF = testGroup "Thompson's group F" + [ testProperty "identity element" prop_identity + , testProperty "associativity" prop_assoc + , testProperty "standard relations" prop_relations + , testProperty "quotient of positives" prop_quot_positive + , testProperty "telescopic product" prop_telescope + , testProperty "cyclic telescopic product (3)" prop_cyclic_product_3 + , testProperty "cyclic telescopic product (4)" prop_cyclic_product_4 + , testProperty "positive diagrams form a monoid" prop_positive_product + , testProperty "composition commutes with reduction" prop_reduce_composition + , testProperty "inverse commutes with reduction" prop_reduce_inverse + ] + +-------------------------------------------------------------------------------- +-- * properties + +prop_relations :: Bool +prop_relations = and [ rel k n | n<-[1..30] , k<-[0..n-1] ] where + rel k n = (inverse $ xk k) `compose` (xk n) `compose` (xk k) == xk (n+1) + +prop_quot_positive :: TPair -> Bool +prop_quot_positive (TPair t1 t2) = (mkTDiag t1 t2) == (positive t1 // positive t2) + +prop_identity :: TDiag -> Bool +prop_identity x = (x ** identity) == x && (identity ** x) == x + +prop_assoc :: TDiag -> TDiag -> TDiag -> Bool +prop_assoc a b c = (p == q) where + p = compose (compose a b) c + q = compose a (compose b c) + +prop_telescope :: TDiag -> TDiag -> TDiag -> TDiag -> Bool +prop_telescope u v w z = (a `compose` b `compose` c) == (u // z) where + a = u // v + b = v // w + c = w // z + +prop_cyclic_product_3 :: TDiag -> TDiag -> TDiag -> Bool +prop_cyclic_product_3 u v w = (a `compose` b `compose` c) == identity where + a = u // v + b = v // w + c = w // u + +prop_cyclic_product_4 :: TDiag -> TDiag -> TDiag -> TDiag -> Bool +prop_cyclic_product_4 u v w z = (a `compose` b `compose` c `compose` d) == identity where + a = u // v + b = v // w + c = w // z + d = z // u + +prop_positive_product :: T -> T -> Bool +prop_positive_product x y = isPositive (positive x `compose` positive y) + +prop_reduce_composition :: Unreduced -> Unreduced -> Bool +prop_reduce_composition (Unreduced x) (Unreduced y) = lhs == rhs where + lhs = reduce (x `composeDontReduce` y) + rhs = compose (reduce x) (reduce y) + +prop_reduce_inverse :: Unreduced -> Bool +prop_reduce_inverse (Unreduced x) = lhs == rhs where + lhs = reduce (inverse x) + rhs = inverse (reduce x) + +-------------------------------------------------------------------------------- +