Unimodality and certain bivariate formal Laurent series

Abstract

In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the operations of addition and multiplication of formal Laurent series. This result is used to establish the unimodality of sequences involving Gauss polynomials and certain refined color partitions. In particular, we solve an open problem posed by Andrews on the unimodality of generalized Gauss polynomials, establish an unimodal result for a statistic of plane partitions, and establish many unimodal results for rank statistics in partition theory, including the rank statistics of concave and convex compositions studied by Andrews, as well as certain unimodal sequences studied by Kim–Lim–Lovejoy. Additionally, we establish the unimodality of the Betti numbers and Gromov–Witten invariants of certain Hilbert schemes of points.