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

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 + n 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 [ m + n 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 [ m + n n ] ‾ q , t . [ABSTRACT FROM AUTHOR]