Alternatingly increasing property and bi-gamma-positivity of polynomials

Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh

Published 2019-07-30Version 1

A polynomial p(z) of degree d is alternatingly increasing if and only if it can be decomposed into a sum p(z)=a(z)+zb(z), where a(z) and b(z) are symmetric and unimodal polynomials with dega(z)=d and degb(z)<d. We say that p(z) is bi-gamma-positive if a(z) and b(z) are both gamma positive. In this paper, we present a unified elementary proof of the bi-gamma-positivity of several polynomials that appear often in algebraic, topological and geometric combinatorics, including q-Eulerian polynomials of types A and B, the descent polynomials of multipermutations and signed multipermutations, Eulerian and derangement polynomials of colored permutations. As an application, we get that these polynomials are all alternatingly increasing.