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

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 barq,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 barq,t. (C) 2018 Elsevier Inc. All rights reserved.