```
-- | 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
--
-- <<svg/young_tableau.svg>>
--
--   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.Integer
import Math.Combinat.Partitions.Integer.IntList ( _dualPartition )
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.
--   <http://www.math.lsa.umich.edu/~jrs/software/SFexamples/tableaux>.
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

--------------------------------------------------------------------------------

```