Math.Combinat.Tableaux.Kostka

This module contains a function to generate (equivalence classes of) triangular tableaux of size k, strictly increasing to the right and to the bottom. For example

  1  
  2  4  
  3  5  8  
  6  7  9  10 

is such a tableau of size 4. The numbers filling a tableau always consist of an interval [1..c]; c is called the content of the tableaux. There is a unique tableau of minimal content 2k-1:

  1  
  2  3  
  3  4  5 
  4  5  6  7 

Let us call the tableaux with maximal content (that is, m = binomial (k+1) 2) standard. The number of standard tableaux are

 1, 1, 2, 12, 286, 33592, 23178480, ...

OEIS:A003121, "Strict sense ballot numbers", http://www.research.att.com/~njas/sequences/A003121.

See R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.

The number of tableaux with content c=m-d are

  d=  |     0      1      2      3    ...
 -----+----------------------------------------------
  k=2 |     1
  k=3 |     2      1
  k=4 |    12     18      8      1
  k=5 |   286    858   1001    572    165     22     1
  k=6 | 33592 167960 361114 436696 326196 155584 47320 8892 962 52 1 

We call these "Kostka tableaux" (in the lack of a better name), since they are in bijection with the simplicial cones in a canonical simplicial decompositions of the Gelfand-Tsetlin cones (the content corresponds to the dimension), which encode the combinatorics of Kostka numbers.

Documentation

type Tableau a = [[a]]

newtype Tri

Set of (i,j) pairs with i>=j>=1.

Constructors

Tri

Fields

unTri :: (Int, Int)

Instances

Eq Tri

Ord Tri

Show Tri

Ix Tri

type TriangularArray a = Array Tri a

Triangular arrays

fromTriangularArray :: TriangularArray a -> Tableau a

triangularArrayUnsafe :: Tableau a -> TriangularArray a

kostkaTableaux :: Int -> [TriangularArray Int]

Generates all tableaux of size k. Effective for k<=6.

_kostkaTableaux :: Int -> [Tableau Int]

countKostkaTableaux :: Int -> [Int]

kostkaContent :: TriangularArray Int -> Int

_kostkaContent :: Tableau Int -> Int