Kostka number

In mathematics, a Kostka number K_λμ, introduced by Kostka (1882), is a non-negative integer depending on two partitions 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 polynomials s_λ as a linear combination of monomial symmetric functions m_μ:

$s_{\lambda }=\sum _{\mu }K_{\lambda \mu }m_{\mu }.\$

Kostka numbers are generalized by the 1 or 2 variable Kostka polynomials.

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.

Kostka (1882, pages 118-120) gave tables of these numbers for partitions of numbers up to 8.

References

• Kostka, C. (1882), "Über den Zusammenhang zwischen einigen Formen von symmetrischen Funktionen", Crelle's J., 93: 89–123
• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR1354144
• Sagan, Bruce E. (2001) [1994], "Schur functions in algebraic combinatorics", Encyclopedia of Mathematics, EMS Press