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

2017/01/27 by Dousse, Jehanne, Kim, Byungchan
#05A10 #05A17 #05A20 #05A30 #11B65 #11P81 #11P84 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1701.07915

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+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 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 m+n \brack n_q,t.

Related

• An overpartition analogue of the q-binomial coefficients
• Combinatorial identities associated with a bivariate generating function for overpartition pairs
• A generalization of the Gaussian formula and a q-analog of Fleck's congruence
• Multi-dimensional q-summations and multi-colored partitions
• (q,t)-analogues and GL_n(F_q)