Quasipolynomials and Maximal Coefficients of Gaussian Polynomials

Authors

Angelica Castillo, The University of Texas Rio Grande Valley
Stephanie Flores, The University of Texas Rio Grande Valley
Anabel Hernandez, The University of Texas Rio Grande Valley
Brandt Kronholm, The University of Texas Rio Grande Valley
Acadia Larsen, The University of Texas Rio Grande Valley
Arturo Martinez, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

2019

Abstract

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 N+m\brack m whose coefficients are precisely p(n, m, N). We give complete characterizations of the maximal coefficients of {M\brack 3} and {M\brack 4}. 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).

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Recommended Citation

Castillo, Angelica, Stephanie Flores, Anabel Hernandez, Brandt Kronholm, Acadia Larsen, and Arturo Martinez. "Quasipolynomials and maximal coefficients of Gaussian polynomials." Annals of Combinatorics 23, no. 3-4 (2019): 589-611. https://doi.org/10.1007/s00026-019-00467-2

Publication Title

Annals of Combinatorics

DOI

10.1007/s00026-019-00467-2