combinat 0.2.6.2 → 0.2.7.0

raw patch · 26 files changed

+1297/−535 lines, 26 filesPVP: major bump suggested

API removals or changes: PVP suggests a major version bump

API changes (from Hackage documentation)

Files

• Math/Combinat.hs +7/−2
• Math/Combinat/ASCII.hs +296/−0
• Math/Combinat/Graphviz.hs +0/−115
• Math/Combinat/Helper.hs +1/−1
• Math/Combinat/LatticePaths.hs +70/−3
• Math/Combinat/Numbers.hs +1/−1
• Math/Combinat/Partitions.hs +3/−331
• Math/Combinat/Partitions/Integer.hs +334/−0
• Math/Combinat/Partitions/Multiset.hs +24/−0
• Math/Combinat/Partitions/NonCrossing.hs +10/−0
• Math/Combinat/Partitions/Plane.hs +116/−0
• Math/Combinat/Partitions/Vector.hs +82/−0
• Math/Combinat/Tableaux.hs +3/−1
• Math/Combinat/Trees/Binary.hs +48/−26
• Math/Combinat/Trees/Binary.hs-boot +22/−0
• Math/Combinat/Trees/Graphviz.hs +115/−0
• Math/Combinat/Trees/Nary.hs +42/−50
• Math/Combinat/Trees/Nary.hs-boot +16/−0
• combinat.cabal +17/−5
• svg/bintrees.svg +4/−0
• svg/dyck_path.svg +4/−0
• svg/ferrers.svg +4/−0
• svg/noncrossing.svg +4/−0
• svg/plane_partition.svg +4/−0
• svg/src/gen_figures.hs +66/−0
• svg/young_tableau.svg +4/−0

@@ -3,7 +3,12 @@ -- objects like partitions, compositions, permutations, -- Young tableaux, various trees, etc etc. --+-- +-- See also the @combinat-diagrams@ library for generating+-- graphical representations of these structure using +-- the @diagrams@ library (<http://projects.haskell.org/diagrams>). --+-- -- The long-term goals are  -- --  (1) generate most of the standard structures;@@ -48,7 +53,7 @@   , module Math.Combinat.Tableaux   , module Math.Combinat.Trees   , module Math.Combinat.LatticePaths-  , module Math.Combinat.Graphviz+  , module Math.Combinat.ASCII   ) where  import Math.Combinat.Numbers@@ -60,4 +65,4 @@ import Math.Combinat.Tableaux import Math.Combinat.Trees import Math.Combinat.LatticePaths-import Math.Combinat.Graphviz+import Math.Combinat.ASCII

@@ -0,0 +1,296 @@+
+-- | A mini-DSL for ASCII drawing of structures.
+--
+--
+-- From some structures there is also Graphviz and\/or @diagrams@ 
+-- (<http://projects.haskell.org/diagrams>) visualization support 
+-- (the latter in the separate libray @combinat-diagrams@).
+--
+
+module Math.Combinat.ASCII where
+
+--------------------------------------------------------------------------------
+
+import Data.List
+
+import Math.Combinat.Helper
+
+--------------------------------------------------------------------------------
+-- * The basic 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]
+  }
+
+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
+                                        
+--------------------------------------------------------------------------------
+-- * Extension
+
+-- | Extends an ASCII figure with spaces horizontally to the given width 
+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
+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
+
+--------------------------------------------------------------------------------
+-- * 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
+
+--------------------------------------------------------------------------------
+-- * 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 
+
+--------------------------------------------------------------------------------
+-- * Concatenation
+
+-- | 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)
+
+-- | 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
+
+--------------------------------------------------------------------------------
+-- * Tabulate
+
+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
+
+--------------------------------------------------------------------------------
+-- * Testing \/ miscellanea
+
+-- | An ASCII 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 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) '-' ++ "/"
+
+asciiNumber :: Int -> ASCII
+asciiNumber = asciiShow
+
+asciiShow :: Show a => a -> ASCII
+asciiShow = asciiFromLines . (:[]) . show
+
+--------------------------------------------------------------------------------

@@ -1,115 +0,0 @@---- | Creates graphviz @.dot@ files from various structures, for example trees.--module Math.Combinat.Graphviz -  ( Dot-  , binTreeDot-  , binTree'Dot-  , treeDot-  , forestDot-  )-  where------------------------------------------------------------------------------------import Data.Tree--import Control.Applicative--import Math.Combinat.Trees.Binary (BinTree(..), BinTree'(..))-import 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"-  -----------------------------------------------------------------------------------binTreeDot :: Show a => String -> BinTree a -> Dot-binTreeDot graphname tree = -  digraphBracket graphname $ binTreeDot' tree--binTree'Dot :: (Show a, Show b) => String -> BinTree' a b -> Dot-binTree'Dot 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.-forestDot -  :: Show a -  => Bool        -- ^ make the individual trees clustered subgraphs-  -> Bool        -- ^ reverse the direction of the arrows-  -> String      -- ^ name of the graph-  -> Forest a -  -> Dot-forestDot 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.-treeDot -  :: Show a -  => Bool     -- ^ reverse the direction of the arrow-  -> String   -- ^ name of the graph-  -> Tree a -  -> Dot-treeDot 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----------------------------------------------------------------------------------

@@ -93,7 +93,7 @@   go b (x : xs) = f b False x : go False xs  ----------------------------------------------------------------------------------- * helpers for ASCII drawing+-- * older helpers for ASCII drawing  -- | extend lines with spaces so that they have the same line mkLinesUniformWidth :: [String] -> [String]

@@ -1,15 +1,23 @@ 
 -- | 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:
+--
+-- <<svg/dyck_path.svg>>
+--
 
 {-# LANGUAGE BangPatterns #-}
 module Math.Combinat.LatticePaths where
 
 --------------------------------------------------------------------------------
 
+import Data.List
 import System.Random
 
 import Math.Combinat.Numbers
 import Math.Combinat.Trees.Binary
+import Math.Combinat.ASCII as ASCII
 
 --------------------------------------------------------------------------------
 -- * Types
@@ -29,7 +37,30 @@ 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 -> '\\' 
+
+--------------------------------------------------------------------------------
+-- * elementary queries
+
 -- | A lattice path is called \"valid\", if it never goes below the @y=0@ line.
 isValidPath :: LatticePath -> Bool
 isValidPath = go 0 where
@@ -71,6 +102,25 @@                         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 !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
@@ -94,7 +144,6 @@                              cnt' = if y'==h then cnt+1 else cnt
                          in  go cnt' (x+1) y' ts
 
-
 --------------------------------------------------------------------------------
 -- * Dyck paths
 
@@ -204,6 +253,12 @@       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
 
@@ -211,7 +266,7 @@ -- 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 touches
+  :: Int   -- ^ @k@ = number of zero-touches
   -> Int   -- ^ @m@ = half-length
   -> [LatticePath]
 touchingDyckPaths = touchingDyckPathsNaive
@@ -226,7 +281,7 @@ -- Naive recursive algorithm, resulting order is pretty ad-hoc.
 --
 touchingDyckPathsNaive
-  :: Int   -- ^ @k@ = number of touches
+  :: Int   -- ^ @k@ = number of zero-touches
   -> Int   -- ^ @m@ = half-length
   -> [LatticePath]
 touchingDyckPathsNaive = worker where
@@ -238,6 +293,18 @@     | 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

@@ -81,7 +81,7 @@ catalanTriangle n k   | k > n     = 0   | k < 0     = 0-  | otherwise = binomial (n+k) n * fromIntegral (n-k+1) `div` fromIntegral (n+1)+  | otherwise = (binomial (n+k) n * fromIntegral (n-k+1)) `div` fromIntegral (n+1)  -------------------------------------------------------------------------------- -- * Stirling numbers

@@ -11,340 +11,12 @@  {-# LANGUAGE BangPatterns #-} module Math.Combinat.Partitions-  ( -- * Type and basic stuff-    Partition-  , toPartition-  , toPartitionUnsafe-  , mkPartition-  , isPartition-  , fromPartition-  , height-  , width-  , heightWidth-  , weight-  , dualPartition-  , _dualPartition-  , elements-  , _elements-  , countAutomorphisms-  , _countAutomorphisms-  , HasNumberOfParts(..)-    -- * Generation-  , partitions'  -  , _partitions' -  , countPartitions'-  , partitions-  , _partitions-  , countPartitions-  , partitionsWithKParts-  , allPartitions'  -  , allPartitions -  , countAllPartitions'-  , countAllPartitions-  , countPartitionsWithKParts-    -- * Ferrer diagrams-  , printFerrerDiagram -  , ferrerDiagram -  , ferrerDiagramEnglishNotation -  , ferrerDiagramFrenchNotation -  , ferrerDiagramEnglishNotation'-  , ferrerDiagramFrenchNotation'-    -- * Paritions of multisets, vector partitions-  , partitionMultiset-  , IntVector-  , vectorPartitions-  , _vectorPartitions-  , fasc3B_algorithm_M-  ) +  ( module Math.Combinat.Partitions.Integer+  )   where -import Data.List-import Data.Array.Unboxed--import Math.Combinat.Helper-import Math.Combinat.Numbers (factorial,binomial,multinomial)- -------------------------------------------------------------------------------- --- | A partition of an integer. The additional invariant enforced here is that partitions ---   are monotone decreasing sequences of positive integers.-newtype Partition = Partition [Int] deriving (Eq,Ord,Show,Read)---- | 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"-  --- | Note: we only check that the sequence is ordered, but we /do not/ check for--- negative elements. This can be useful when working with symmetric functions.--- It may also change in the future...-isPartition :: [Int] -> Bool-isPartition []  = True-isPartition [_] = True-isPartition (x:xs@(y:_)) = (x >= y) && isPartition xs--fromPartition :: Partition -> [Int]-fromPartition (Partition part) = part---- | The first element of the sequence.-height :: Partition -> Int-height (Partition part) = case part of-  (p:_) -> p-  [] -> 0-  --- | The length of the sequence.-width :: Partition -> Int-width (Partition part) = length part--heightWidth :: Partition -> (Int,Int)-heightWidth part = (height part, width part)---- | The weight of the partition ---   (that is, the sum of the corresponding sequence).-weight :: Partition -> Int-weight (Partition part) = sum' part---- | The dual (or conjugate) partition.-dualPartition :: Partition -> Partition-dualPartition (Partition part) = Partition (_dualPartition part)---- (we could be more efficient, but it hardly matters)-_dualPartition :: [Int] -> [Int]-_dualPartition [] = []-_dualPartition xs@(k:_) = [ length $ filter (>=i) xs | i <- [1..k] ]---- | 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] ] ---- | Computes the number of \"automorphisms\" of a given integer partition.-countAutomorphisms :: Partition -> Integer  -countAutomorphisms = _countAutomorphisms . fromPartition--_countAutomorphisms :: [Int] -> Integer-_countAutomorphisms = multinomial . map length . group-------------------------------------------------------------------------------------class HasNumberOfParts p where-  numberOfParts :: p -> Int--instance HasNumberOfParts Partition where-  numberOfParts (Partition p) = length p-  -------------------------------------------------------------------------------------- | 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] ] ---- | Partitions of @d@, as lists-_partitions :: Int -> [[Int]]-_partitions d = _partitions' (d,d) d---- | Partitions of @d@.-partitions :: Int -> [Partition]-partitions d = partitions' (d,d) d--countPartitions :: Int -> Integer-countPartitions d = countPartitions' (d,d) d---- | All integer partitions fitting into a given rectangle.-allPartitions'  -  :: (Int,Int)        -- ^ (height,width)-  -> [[Partition]]-allPartitions' (h,w) = [ partitions' (h,w) i | i <- [0..d] ] where d = h*w---- | 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 = [ partitions i | i <- [0..d] ]---- | # = \\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] ]------------------------------------------------------------------------------------- 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))] ]-------------------------------------------------------------------------------------- * Ferrer diagrams--printFerrerDiagram :: Partition -> IO ()-printFerrerDiagram = putStrLn . ferrerDiagram---- | Synonym for 'ferrerDiagramEnglishNotation'-ferrerDiagram :: Partition -> String-ferrerDiagram = ferrerDiagramEnglishNotation--ferrerDiagramEnglishNotation :: Partition -> String-ferrerDiagramEnglishNotation = ferrerDiagramEnglishNotation' '@'--ferrerDiagramFrenchNotation :: Partition -> String-ferrerDiagramFrenchNotation  = ferrerDiagramFrenchNotation'  '@'--ferrerDiagramEnglishNotation' :: Char -> Partition -> String-ferrerDiagramEnglishNotation' ch part = unlines (map f ys) where-  ys  = fromPartition part-  f n = replicate n ch --ferrerDiagramFrenchNotation' :: Char -> Partition -> String-ferrerDiagramFrenchNotation' ch part = unlines (map f ys) where-  ys  = reverse $ fromPartition $ dualPartition part-  f n = replicate n ch -------------------------------------------------------------------------------------- | Partitions of a multiset.-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---- | 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+import Math.Combinat.Partitions.Integer  --------------------------------------------------------------------------------

@@ -0,0 +1,334 @@++-- | 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.+--+--  * <http://en.wikipedia.org/wiki/Partition_(number_theory)>+--+-- For example the partition+--+-- > Partition [8,6,3,3,1]+--+-- can be represented by the (English notation) Ferrers diagram:+--+-- <<svg/ferrers.svg>>+-- ++{-# LANGUAGE BangPatterns #-}+module Math.Combinat.Partitions.Integer where++--------------------------------------------------------------------------------++import Data.List++import Math.Combinat.Helper+import Math.Combinat.ASCII as ASCII+import Math.Combinat.Numbers (factorial,binomial,multinomial)++--------------------------------------------------------------------------------+-- * 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)++---------------------------------------------------------------------------------++class HasNumberOfParts p where+  numberOfParts :: p -> Int++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"+  +-- | Note: we only check that the sequence is ordered, but we /do not/ check for+-- negative elements. This can be useful when working with symmetric functions.+-- It may also change in the future...+isPartition :: [Int] -> Bool+isPartition []  = True+isPartition [_] = True+isPartition (x:xs@(y:_)) = (x >= y) && isPartition xs++fromPartition :: Partition -> [Int]+fromPartition (Partition part) = part++-- | The first element of the sequence.+height :: Partition -> Int+height (Partition part) = case part of+  (p:_) -> p+  [] -> 0+  +-- | The length of the sequence.+width :: Partition -> Int+width (Partition part) = length part++heightWidth :: Partition -> (Int,Int)+heightWidth part = (height part, width part)++-- | The weight of the partition +--   (that is, the sum of the corresponding sequence).+weight :: Partition -> Int+weight (Partition part) = sum' part++-- | The dual (or conjugate) partition.+dualPartition :: Partition -> Partition+dualPartition (Partition part) = Partition (_dualPartition part)++-- (we could be more efficient, but it hardly matters)+_dualPartition :: [Int] -> [Int]+_dualPartition [] = []+_dualPartition xs@(k:_) = [ length $ filter (>=i) xs | i <- [1..k] ]++-- | 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] ] ++---------------------------------------------------------------------------------+-- * 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++---------------------------------------------------------------------------------+-- * 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 <http://en.wikipedia.org/wiki/Dominance_order>+--+dominates :: Partition -> Partition -> Bool+dominates (Partition qs) (Partition ps) +  = and $ zipWith (>=) (sums (qs ++ repeat 0)) (sums ps)+  where+    sums = scanl (+) 0++---------------------------------------------------------------------------------+-- * Generating partitions++-- | 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] ] ++-- | Partitions of @d@, as lists+_partitions :: Int -> [[Int]]+_partitions d = _partitions' (d,d) d++-- | Partitions of @d@.+partitions :: Int -> [Partition]+partitions d = partitions' (d,d) d++countPartitions :: Int -> Integer+countPartitions d = countPartitions' (d,d) d++-- | All integer partitions fitting into a given rectangle.+allPartitions'  +  :: (Int,Int)        -- ^ (height,width)+  -> [[Partition]]+allPartitions' (h,w) = [ partitions' (h,w) i | i <- [0..d] ] where d = h*w++-- | 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 = [ partitions i | i <- [0..d] ]++-- | # = \\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] ]++--------------------------------------------------------------------------------+-- * 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))] ]+++--------------------------------------------------------------------------------+-- * Sub-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)++----------------------------------------++-- | 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 ]++--------------------------------------------------------------------------------+-- * 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++--------------------------------------------------------------------------------

@@ -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
+
+--------------------------------------------------------------------------------

@@ -3,6 +3,16 @@ --
 -- See eg. <http://en.wikipedia.org/wiki/Noncrossing_partition>
 --
+-- 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
+--
+-- <<svg/noncrossing.svg>>
+--
+-- is represented as
+--
+-- > NonCrossing [[3],[5,4,2],[7,6,1],[9,8]]
+--
 
 {-# LANGUAGE BangPatterns #-}
 module Math.Combinat.Partitions.NonCrossing where

@@ -0,0 +1,116 @@+
+-- | Plane partitions. See eg. <http://en.wikipedia.org/wiki/Plane_partition>
+--
+-- Plane partitions are encoded as lists of lists of Z heights. For example the plane 
+-- partition in the picture
+-- 
+-- <<svg/plane_partition.svg>>
+--
+-- 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.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)
+
+-- | 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.shape . 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)
+
+--------------------------------------------------------------------------------
+-- * 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 ]
+
+--------------------------------------------------------------------------------

@@ -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
+
+--------------------------------------------------------------------------------

@@ -6,11 +6,13 @@ --   The convention is that we use  --   the English notation, and we store the tableaux as lists of the rows. -- ---   That is, the following standard tableau of shape [5,4,1]+--   That is, the following standard Young tableau of shape [5,4,1] --  -- >  1  3  4  6  7 -- >  2  5  8 10 -- >  9+--+-- <<svg/young_tableau.svg>> -- --   is encoded conveniently as -- 

@@ -1,27 +1,24 @@  -- | 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:+--+-- <<svg/bintrees.svg>>+--  module Math.Combinat.Trees.Binary    ( -- * Types     BinTree(..)   , leaf   , BinTree'(..)-  , toRoseTree , toRoseTree'   , forgetNodeDecorations-  , module Data.Tree    , Paren(..)   , parenthesesToString   , stringToParentheses-    -- * Bijections-  , forestToNestedParentheses-  , forestToBinaryTree-  , nestedParenthesesToForest-  , nestedParenthesesToForestUnsafe-  , nestedParenthesesToBinaryTree-  , nestedParenthesesToBinaryTreeUnsafe-  , binaryTreeToForest-  , binaryTreeToNestedParentheses+  -- * Conversion to rose trees (@Data.Tree@)+  , toRoseTree , toRoseTree'+  , module Data.Tree      -- * Nested parentheses   , nestedParentheses    , randomNestedParentheses@@ -30,15 +27,29 @@   , fasc4A_algorithm_P   , fasc4A_algorithm_W   , fasc4A_algorithm_U-    -- * Binary trees+    -- * Generating binary trees   , binaryTrees   , countBinaryTrees   , binaryTreesNaive   , randomBinaryTree   , fasc4A_algorithm_R     -- * ASCII drawing-  , printBinaryTree_-  , drawBinaryTree_+  , asciiBinaryTree_+    -- * Graphviz drawing+  , Dot+  , graphvizDotBinTree+  , graphvizDotBinTree'+  , graphvizDotForest+  , graphvizDotTree  +    -- * Bijections+  , forestToNestedParentheses+  , forestToBinaryTree+  , nestedParenthesesToForest+  , nestedParenthesesToForestUnsafe+  , nestedParenthesesToBinaryTree+  , nestedParenthesesToBinaryTreeUnsafe+  , binaryTreeToForest+  , binaryTreeToNestedParentheses   )    where @@ -61,9 +72,17 @@  import System.Random -import Math.Combinat.Helper import Math.Combinat.Numbers (factorial,binomial) +import Math.Combinat.Trees.Graphviz +  ( Dot +  , graphvizDotBinTree , graphvizDotBinTree' +  , graphvizDotForest , graphvizDotTree +  )++import Math.Combinat.Helper+import Math.Combinat.ASCII as ASCII+ -------------------------------------------------------------------------------- -- * Types @@ -89,7 +108,7 @@ forgetNodeDecorations (Leaf' decor) = Leaf decor   ----------------------------------------------------------------------------------- * conversion to Data.Tree+-- * conversion to 'Data.Tree'  -- | Convert a binary tree to a rose tree (from "Data.Tree") toRoseTree :: BinTree a -> Tree (Maybe a)@@ -103,7 +122,7 @@   go (Leaf' x)         = Node (Right x) []     ----------------------------------------------------------------------------------- * instances+-- instances    instance Functor BinTree where   fmap f = go where@@ -121,9 +140,12 @@     go (Branch left right) = Branch <$> go left <*> go right  ----------------------------------------------------------------------------------- * nester parentheses+-- * Nested parentheses -data Paren = LeftParen | RightParen deriving (Eq,Ord,Show,Read)+data Paren +  = LeftParen +  | RightParen +  deriving (Eq,Ord,Show,Read)  parenToChar :: Paren -> Char parenToChar LeftParen = '('@@ -299,7 +321,7 @@       c' = ((q+1)*(q-p)*c) `div` ((q+p)*(q-p+1))    ----------------------------------------------------------------------------------- * Binary trees+-- * Generating binary trees  -- | Generates all binary trees with n nodes.  --   At the moment just a synonym for 'binaryTreesNaive'.@@ -359,18 +381,16 @@       (k,b) = x `divMod` 2        --------------------------------------------------------------------------------      +-- * ASCII drawing    -- | Draws a binary tree in ASCII, ignoring node labels. -- -- Example: ----- > mapM_ printBinaryTree_ $ binaryTrees 4+-- > autoTabulate RowMajor (Right 5) $ map asciiBinaryTree_ $ binaryTrees 4 ---printBinaryTree_ :: BinTree a -> IO ()-printBinaryTree_ = putStrLn . drawBinaryTree_-  -drawBinaryTree_ :: BinTree a -> String-drawBinaryTree_ = unlines . fst . go where+asciiBinaryTree_ :: BinTree a -> ASCII+asciiBinaryTree_ = ASCII.asciiFromLines . fst . go where    go :: BinTree a -> ([String],Int)   go (Leaf x) = ([],0)@@ -389,3 +409,5 @@   blockWidth ls = case ls of     (l:_) -> length l     []    -> 0++--------------------------------------------------------------------------------      

@@ -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++--------------------------------------------------------------------------------

@@ -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++--------------------------------------------------------------------------------

@@ -2,47 +2,48 @@ -- | N-ary trees.  module Math.Combinat.Trees.Nary -  (-    -- * regular trees +  (      +    -- * Regular trees      ternaryTrees   , regularNaryTrees   , semiRegularTrees   , countTernaryTrees   , countRegularNaryTrees-    -- * derivation trees+    -- * \"derivation trees\"   , derivTrees     -- * ASCII drawings-  , printTreeVertical_-  , printTreeVertical-  , printTreeVerticalLeavesOnly-  , drawTreeVertical_-  , drawTreeVertical-  , drawTreeVerticalLeavesOnly-    -- * classifying nodes+  , asciiTreeVertical_+  , asciiTreeVertical+  , asciiTreeVerticalLeavesOnly+    -- * Graphviz drawing+  , Dot+  , graphvizDotTree  +  , graphvizDotForest+    -- * Classifying nodes   , classifyTreeNode   , isTreeLeaf  , isTreeNode   , isTreeLeaf_ , isTreeNode_   , treeNodeNumberOfChildren -    -- * counting nodes+    -- * Counting nodes   , countTreeNodes   , countTreeLeaves   , countTreeLabelsWith   , countTreeNodesWith -    -- * left and right spines+    -- * Left and right spines   , leftSpine  , leftSpine_   , rightSpine , rightSpine_   , leftSpineLength , rightSpineLength-    -- * unique labels+    -- * Unique labels   , addUniqueLabelsTree   , addUniqueLabelsForest   , addUniqueLabelsTree_   , addUniqueLabelsForest_-    -- * labelling by depth+    -- * Labelling by depth   , labelDepthTree   , labelDepthForest   , labelDepthTree_   , labelDepthForest_-    -- * labelling by number of children+    -- * Labelling by number of children   , labelNChildrenTree   , labelNChildrenForest   , labelNChildrenTree_@@ -62,11 +63,14 @@ import Control.Monad.Trans.State import Data.Traversable (traverse) -import Math.Combinat.Sets         (listTensor)-import Math.Combinat.Partitions   (partitionMultiset)-import Math.Combinat.Compositions (compositions)-import Math.Combinat.Numbers      (factorial,binomial)+import Math.Combinat.Sets                  ( listTensor )+import Math.Combinat.Partitions.Multiset   ( partitionMultiset )+import Math.Combinat.Compositions          ( compositions )+import Math.Combinat.Numbers               ( factorial, binomial ) +import Math.Combinat.Trees.Graphviz ( Dot , graphvizDotForest , graphvizDotTree )++import Math.Combinat.ASCII as ASCII import Math.Combinat.Helper  --------------------------------------------------------------------------------@@ -107,9 +111,9 @@ -- | All trees on @n@ nodes where the number of children of all nodes is -- in element of the given set. Example: ----- > mapM_ printTreeVertical --- >  $ map labelNChildrenTree_ --- >  $ semiRegularTrees [2,3] n+-- > 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,...] --@@ -160,31 +164,14 @@               -------------------------------------------------------------------------------- --- | Vertical ASCII drawing of a tree, without labels.--- --- Example:------ > mapM_ printTreeVertical_ $ regularNaryTrees 2 3 ----printTreeVertical_ :: Tree a -> IO ()-printTreeVertical_ = putStrLn . drawTreeVertical_---- | Prints all labels.+-- | Vertical ASCII drawing of a tree, without labels. Example: ----- Example: +-- > autoTabulate RowMajor (Right 5) $ map asciiTreeVertical_ $ regularNaryTrees 2 4  ----- > printTreeVertical $ addUniqueLabelsTree_ $ (regularNaryTrees 3 9) !! 666+-- Nodes are denoted by @\@@, leaves by @*@. ---printTreeVertical :: Show a => Tree a -> IO ()-printTreeVertical = putStrLn . drawTreeVertical---- | Prints the labels for the leaves, but not for the nonempty nodes-printTreeVerticalLeavesOnly :: Show a => Tree a -> IO ()-printTreeVerticalLeavesOnly = putStrLn . drawTreeVerticalLeavesOnly---- | Nodes are denoted by @\@@, leaves by @*@.-drawTreeVertical_ :: Tree a -> String-drawTreeVertical_ tree = unlines (go tree) where+asciiTreeVertical_ :: Tree a -> ASCII+asciiTreeVertical_ tree = ASCII.asciiFromLines (go tree) where   go :: Tree b -> [String]   go (Node _ cs) = case cs of     [] -> ["-*"]@@ -199,9 +186,14 @@                                              else "+-"                    in  (branch++l) : map (indent++) ls ++ gap --- | Nodes are denoted by @(label)@, leaves by @label@.-drawTreeVertical :: Show a => Tree a -> String-drawTreeVertical tree = unlines (go tree) where+-- | 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]@@ -220,9 +212,9 @@                              else " +" ++ dashes ++ "--"         in  (branch++l) : map (indent++) ls ++ gap --- | Nodes are denoted by @\@@, leaves by @label@.-drawTreeVerticalLeavesOnly :: Show a => Tree a -> String-drawTreeVerticalLeavesOnly tree = unlines (go tree) where+-- | 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]

@@ -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++--------------------------------------------------------------------------------

@@ -1,5 +1,5 @@ Name:                combinat-Version:             0.2.6.2+Version:             0.2.7.0 Synopsis:            Generation of various combinatorial objects. Description:         A collection of functions to generate (and if there is                       a formula, count) combinatorial objects like partitions, @@ -14,9 +14,14 @@ Stability:           Experimental Category:            Math Tested-With:         GHC == 7.8.3-Cabal-Version:       >= 1.6+Cabal-Version:       >= 1.18 Build-Type:          Simple +extra-doc-files:     svg/*.svg ++extra-source-files:  svg/*.svg+                     svg/src/gen_figures.hs                     + Flag withQuickCheck   Description: Compile with the QuickCheck tests.    default: False@@ -36,7 +41,6 @@   if flag(withQuickCheck)     Build-Depends:       QuickCheck -   Exposed-Modules:     Math.Combinat                        Math.Combinat.Numbers                        Math.Combinat.Numbers.Series@@ -45,21 +49,29 @@                        Math.Combinat.Tuples                         Math.Combinat.Compositions                        Math.Combinat.Partitions+                       Math.Combinat.Partitions.Integer                        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.Kostka                        Math.Combinat.Trees                        Math.Combinat.Trees.Binary                        Math.Combinat.Trees.Nary+                       Math.Combinat.Trees.Graphviz                        Math.Combinat.LatticePaths                        Math.Combinat.FreeGroups-                       Math.Combinat.Graphviz+                       Math.Combinat.ASCII                        Math.Combinat.Helper -  Extensions:          CPP, MultiParamTypeClasses, ScopedTypeVariables, +  Default-Extensions:  CPP, BangPatterns+  Other-Extensions:    MultiParamTypeClasses, ScopedTypeVariables,                         GeneralizedNewtypeDeriving, BangPatterns ++  Default-Language:    Haskell2010    Hs-Source-Dirs:      . 

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@@ -0,0 +1,66 @@+
+-- | 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.Tableaux
+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.Tableaux
+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) 
+
+--------------------------------------------------------------------------------
+
+main = do 
+
+  export "plane_partition.svg" (Width 320) $ drawPlanePartition3D $
+    PlanePart [[5,4,3,3,1],[4,4,2,1],[3,2],[2,1],[1],[1]] 
+
+  export "noncrossing.svg" (Width 256) $ pad 1.10 $ drawNonCrossingCircleDiagram' orange True $
+    NonCrossing [[3],[5,4,2],[7,6,1],[9,8]]
+
+  export "young_tableau.svg" (Width 256) $ 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" (Width 500) $ drawLatticePath $ path
+  -- print (pathHeight path, pathNumberOfZeroTouches path, pathNumberOfPeaks path)
+
+  export "ferrers.svg" (Width 256) $ drawFerrersDiagram' EnglishNotation red True $
+    Partition [8,6,3,3,1]
+
+  export "bintrees.svg" (Width 750) $ boxSep 7 $ map drawBinTree_ (binaryTrees 4)
+
+--------------------------------------------------------------------------------

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