Math.Combinat.Tableaux

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 tableau of shape [5,4,1]

 1  3  4  6  7
 2  5  8 10
 9

is encoded conveniently as

[ [ 1 , 3 , 4 , 6 , 7 ]
, [ 2 , 5 , 8 ,10 ]
, [ 9 ]
]

Basic stuff

type Tableau a = [[a]]

_shape :: Tableau a -> [Int]

shape :: Tableau a -> Partition

dualTableau :: Tableau a -> Tableau a

content :: Tableau a -> [a]

hooks :: Partition -> Tableau (Int, Int)

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)]

hookLengths :: Partition -> Tableau Int

Row and column words

rowWord :: Tableau a -> [a]

rowWordToTableau :: Ord a => [a] -> Tableau a

columnWord :: Tableau a -> [a]

columnWordToTableau :: Ord a => [a] -> Tableau a

Standard Young tableaux

standardYoungTableaux :: Partition -> [Tableau Int]

Standard Young tableaux of a given shape. Adapted from John Stembridge, http://www.math.lsa.umich.edu/~jrs/software/SFexamples/tableaux.

countStandardYoungTableaux :: Partition -> Integer

hook-length formula

Semistandard Young tableaux

semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int]

Semistandard Young tableaux of given shape, "naive" algorithm

countSemiStandardYoungTableaux :: Int -> Partition -> Integer

Stanley's hook formula (cf. Fulton page 55)