Macdonald has defined a two-parameter refinement of the Kostka matrix, denoted KÀyt,(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 ß is a partition with at most two columns (or at most two rows), then Kxn(q,t) is indeed a polynomial. Second, we provide a combinatorial interpretation of Kx^(q, t) for the case in which p

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... is a hook. This interpretation proves in this case that not only are the entries polynomials, but also that their coefficients are nonnegative integers.