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

Abstract

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an m×n rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization [m+nn]‾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+nn]‾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+nn]‾q,t.