combinat-0.2.8.0: Math/Combinat/Partitions/Integer.hs
-- | 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 CPP, BangPatterns #-}
module Math.Combinat.Partitions.Integer where
--------------------------------------------------------------------------------
import Data.List
-- 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
--------------------------------------------------------------------------------
-- * 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) ]
countPartitions :: Int -> Integer
countPartitions d = countPartitions' (d,d) d
-- | 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] ]
---------------------------------------------------------------------------------
-- * 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
-- | 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 <http://en.wikipedia.org/wiki/Pieri's_formula>
--
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
--------------------------------------------------------------------------------
#ifdef QUICKCHECK
-- * Tests
-- we need some custom types for quickcheck generating not too large partitions...
newtype PartitionWeight = PartitionWeight Int
data PartitionWeightPair = PartitionWeightPair Int Int
data PartitionPair = PartitionPair Partition Int
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 :: PartitionPair -> Bool
prop_subparts (PartitionPair 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 ])
#endif
--------------------------------------------------------------------------------