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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1182707-1

MathSciNet review:
1182707

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Abstract: Macdonald has defined a two-parameter refinement of the Kostka matrix, denoted . 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 is a partition with at most two columns (or at most two rows), then is indeed a polynomial. Second, we provide a combinatorial interpretation of for the case in which 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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DOI:
https://doi.org/10.1090/S0002-9939-1994-1182707-1

Article copyright:
© Copyright 1994
American Mathematical Society