On [formula omitted]-analogs of descent and peak polynomials

Abstract

Descent polynomials and peak polynomials, which enumerate permutations π∈Sn with given descent and peak sets respectively, have recently received considerable attention (Billey et al., 1989;Diaz-Lopez et al., 2019). We give several formulas for q-analogs of these polynomials which refine the enumeration by the length of π. In the case of q-descent polynomials we prove that the coefficients in one basis are stronglyq-log concave, and conjecture this property in another basis. For peaks, we prove that the q-peak polynomial is palindromic in q, resolving a conjecture of Diaz-Lopez et al. (2021).