Some particular entries of the two-parameter Kostka matrix

Abstract:

Macdonald has defined a two-parameter refinement of the Kostka matrix, denoted ${K_{\lambda ,\mu }}(q,t)$. The entries are rational functions of q and t, but he has conjectured that they are in fact polynomials with nonnegative integer coefficients. We prove two results that support this conjecture. First, we prove that if $\mu$ is a partition with at most two columns (or at most two rows), then ${K_{\lambda ,\mu }}(q,t)$ is indeed a polynomial. Second, we provide a combinatorial interpretation of ${K_{\lambda ,\mu }}(q,t)$ for the case in which $\mu$ is a hook. This interpretation proves in this case that not only are the entries polynomials, but also that their coefficients are nonnegative integers.