Some particular entries of the two-parameter Kostka matrix

Abstract

Macdonald has defined a two-parameter refinement of the Kostka matrix, denoted K λ , μ ( q , t ) {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 μ \mu is a partition with at most two columns (or at most two rows), then K λ , μ ( q , t ) {K_{\lambda ,\mu }}(q,t) is indeed a polynomial. Second, we provide a combinatorial interpretation of K λ , μ ( q , t ) {K_{\lambda ,\mu }}(q,t) for the case in which μ \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.