Kostka number
In mathematics, the Kostka number
(depending on two integer partitions
) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape
They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Kostka (1882)).
, the Kostka number
counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows.
, the Kostka number
is equal to 1: the unique way to fill the Young diagram of shape
copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on.
(This tableau is sometimes called the Yamanouchi tableau of shape
is positive (i.e., there exist semistandard Young tableaux of shape
[2] In general, there are no nice formulas known for the Kostka numbers.
However, some special cases are known.
is the partition whose parts are all 1 then a semistandard Young tableau of weight
is a standard Young tableau; the number of standard Young tableaux of a given shape
An important simple property of Kostka numbers is that
does not depend on the order of entries of
This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape
differ only by swapping two entries.
[3] In addition to the purely combinatorial definition above, they can also be defined as the coefficients that arise when one expresses the Schur polynomial
as a linear combination of monomial symmetric functions
Alternatively, Schur polynomials can also be expressed[4] as where the sum is over all weak compositions
On the level of representations of the symmetric group
, Kostka numbers express the decomposition of the permutation module
in terms of the irreducible representations
, i.e., On the level of representations of the general linear group
also counts the dimension of the weight space corresponding to
in the unitary irreducible representation
The Kostka numbers for partitions of size at most 3 are as follows: These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions: Kostka (1882, pages 118-120) gave tables of these numbers for partitions of numbers up to 8.
Kostka numbers are special values of the 1 or 2 variable Kostka polynomials:
