In mathematics, a Kostka number K_λμ, introduced by , is a non-negative integer  depending on two partition

Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...

s λ and μ, that is equal to the number of semistandard Young tableaux of shape λ and weight μ.
They can be used to express Schur polynomial

Schur polynomial

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of...

s s_λ as a linear combination

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

 of monomial symmetric functions m_μ:

Kostka numbers also express the decomposition of the permutation module M_μ in terms of the representations V_λ corresponding to the character s_λ, i.e.

On the level of representations of , the Kostka number K_λμ counts the dimension of the weight space corresponding to μ in the irreducible representation V_λ (where we require μ and λ to have at most n parts).

Kostka numbers are special values of the 1 or 2 variable Kostka polynomial

Kostka polynomial

In mathematics, a Kostka polynomial or Kostka–Foulkes polynomial Kλμ, named after Carl Kostka, is a polynomial in two variables with non-negative integer coefficients depending on two partitions λ and μ...

s:

Examples

The Kostka numbers for partitions of size at most 3 are given by the coefficients of:

s = m = 1 (indexed by the empty partition)

s₁ = m₁

s₂ = m₂ + m₁₁

s₁₁ = m₁₁

s₃ = m₃ + m₂₁ + m₁₁₁

s₂₁ = m₂₁ + 2m₁₁₁

s₁₁₁ = m_111.

gave tables of these numbers for partitions of numbers up to 8.