Kostka polynomial

They are studied primarily in algebraic combinatorics and representation theory.

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.

[1] 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.