Young tableaux and similar gadgets.
See e.g. William Fulton: Young Tableaux, with Applications to Representation theory and Geometry (CUP 1997).
The convention is that we use the English notation, and we store the tableaux as lists of the rows.
That is, the following standard Young tableau of shape [5,4,1]
1 3 4 6 7 2 5 8 10 9
is encoded conveniently as
[ [ 1 , 3 , 4 , 6 , 7 ] , [ 2 , 5 , 8 ,10 ] , [ 9 ] ]
- type Tableau a = [[a]]
- asciiTableau :: Show a => Tableau a -> ASCII
- _tableauShape :: Tableau a -> [Int]
- tableauShape :: Tableau a -> Partition
- tableauWeight :: Tableau a -> Int
- dualTableau :: Tableau a -> Tableau a
- tableauContent :: Tableau a -> [a]
- hooks :: Partition -> Tableau (Int, Int)
- hookLengths :: Partition -> Tableau Int
- rowWord :: Tableau a -> [a]
- rowWordToTableau :: Ord a => [a] -> Tableau a
- columnWord :: Tableau a -> [a]
- columnWordToTableau :: Ord a => [a] -> Tableau a
- isLatticeWord :: [Int] -> Bool
- isSemiStandardTableau :: Tableau Int -> Bool
- semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int]
- countSemiStandardYoungTableaux :: Int -> Partition -> Integer
- isStandardTableau :: Tableau Int -> Bool
- standardYoungTableaux :: Partition -> [Tableau Int]
- countStandardYoungTableaux :: Partition -> Integer
The dual of the tableau is the mirror image to the main diagonal.
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
(i,j) of the resulting tableau (which has shape of the
given partition) means that the vertical part of the hook has length
and the horizontal part
j. The hook length is thus
> 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)]
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.
Semistandard tableaux can be reconstructed from their row words
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
Standard tableaux can be reconstructed from either their column or row words
Checks whether a sequence of positive integers is a lattice word,
which means that in every initial part of the sequence any number
occurs at least as often as the number
Semistandard Young tableaux
A tableau is semistandard if its entries are weekly increasing horizontally and strictly increasing vertically
Semistandard Young tableaux of given shape, "naive" algorithm
Stanley's hook formula (cf. Fulton page 55)
Standard Young tableaux
A tableau is standard if it is semistandard and its content is exactly
n is the weight.
Standard Young tableaux of a given shape. Adapted from John Stembridge, http://www.math.lsa.umich.edu/~jrs/software/SFexamples/tableaux.