Safe Haskell | None |
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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 ] ]
- 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)
- hookLengths :: Partition -> Tableau Int
- rowWord :: Tableau a -> [a]
- rowWordToTableau :: Ord a => [a] -> Tableau a
- columnWord :: Tableau a -> [a]
- columnWordToTableau :: Ord a => [a] -> Tableau a
- standardYoungTableaux :: Partition -> [Tableau Int]
- countStandardYoungTableaux :: Partition -> Integer
- semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int]
- countSemiStandardYoungTableaux :: Int -> Partition -> Integer
Basic stuff
dualTableau :: Tableau a -> Tableau aSource
hooks :: Partition -> Tableau (Int, Int)Source
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 IntSource
Row and column words
rowWordToTableau :: Ord a => [a] -> Tableau aSource
columnWord :: Tableau a -> [a]Source
columnWordToTableau :: Ord a => [a] -> Tableau aSource
Standard Young tableaux
standardYoungTableaux :: Partition -> [Tableau Int]Source
Standard Young tableaux of a given shape. Adapted from John Stembridge, http://www.math.lsa.umich.edu/~jrs/software/SFexamples/tableaux.
countStandardYoungTableaux :: Partition -> IntegerSource
hook-length formula
Semistandard Young tableaux
semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int]Source
Semistandard Young tableaux of given shape, "naive" algorithm
countSemiStandardYoungTableaux :: Int -> Partition -> IntegerSource
Stanley's hook formula (cf. Fulton page 55)