An overpartition analogue of q-binomial coefficients, II: Combinatorial proofs and (q, t)-log concavity

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:158

Article

Dousse, Jehanne¹  Kim, Byungchan² 

[1] Univ Zurich, Inst Math, Winterthurerstr 190, CH-8057 Zurich, Switzerland

[2] Seoul Natl Univ Sci & Technol, Sch Liberal Arts, 232 Gongneung Ro, Seoul 01811, South Korea

关键词: Gaussian polynomial;    q-Binomial coefficient;    Over-(q, t)-binomial coefficient;    Finite versions of q-series identities;    Combinatorial proofs;    q-log concavity;    Delannoy numbers;    Overpartitions;   

DOI  :  10.1016/j.jcta.2018.03.011

来源: Elsevier

【 摘 要 】

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an m x n rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization <([(m+n)(n)])over bar>(q,t) of Gaussian polynomials, which is also a (q, t)-analogue of Delannoy numbers. First we obtain finite versions of classical q-series identities such as the q-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite theta function and prove the (q, t)-log concavity of <([(m+n)(n)])over bar>q,t. We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of <([(m+n)(n)])over bar>q,t. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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