In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory. The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t). There are two slightly different versions of them, one called transformed Kostka polynomials. The one-variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials Pμ to Schur polynomials sλ: These polynomials were conjectured to have non-negative integer coefficients by Foulkes, and this was later proved in 1978 by Alain Lascoux and Marcel-Paul Schützenberger. In fact, they show that where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ. Here, charge is a certain combinatorial statistic on semi-standard Young tableaux. The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by Pμ) to Schur polynomials sλ: where Kostka numbers are special values of the one- or two-variable Kostka polynomials:

Examples