## Some particular entries of the two-parameter Kostka matrix

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- by John R. Stembridge PDF
- Proc. Amer. Math. Soc.
**121**(1994), 367-373 Request permission

## 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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## Additional Information

- © Copyright 1994 American Mathematical Society
- 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