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

Autor: Jehanne Dousse, Byungchan Kim
Rok vydání: 2018
Předmět:
Zdroj: Journal of Combinatorial Theory, Series A. 158:228-253
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2018.03.011
Popis: In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{{m+n \brack n}}_{q,t}$ 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 $\overline{{m+n \brack n}}_{q,t}$. 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 $\overline{{m+n \brack n}}_{q,t}$.
Comment: 19 pages, 2 figures, v2: corrected typo
Databáze: OpenAIRE
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