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Quasipolynomials and maximal coefficients of Gaussian polynomials. (English) Zbl 1456.11194

Summary: We establish an algorithm for producing formulas for \(p(n, m, N)\), the function enumerating partitions of \(n\) into at most \(m\) parts with no part larger than \(N\). Recent combinatorial results of H. Hahn et al. on a collection of partition identities for \(p(n, 3, N)\) are considered. We offer direct proofs of these identities and then place them in a larger context of the unimodality of Gaussian polynomials \(\left[ \begin{matrix}N+m \\ m\end{matrix}\right]\) whose coefficients are precisely \(p(n, m, N)\). We give complete characterizations of the maximal coefficients of \(\left[ \begin{matrix} M \\ 3\end{matrix}\right]\) and \(\left[\begin{matrix}M\\ 4\end{matrix}\right]\). Furthermore, we prove a general theorem on the period of quasipolynomials for central/maximal coefficients of Gaussian polynomials. We place some of Hahn’s identities into the context of some known results on differences of partitions into at most \(m\) parts, \(p(n, m)\), which we then extend to \(p(n, m, N)\).

MSC:

11P81 Elementary theory of partitions
05A15 Exact enumeration problems, generating functions

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References:

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