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Some particular entries of the two-parameter Kostka matrix

Author: John R. Stembridge
Journal: Proc. Amer. Math. Soc. 121 (1994), 367-373
MSC: Primary 05E05; Secondary 05E15
MathSciNet review: 1182707
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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.

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Article copyright: © Copyright 1994 American Mathematical Society