A Recurrence Relation for the "inv" Analogue of $q$-Eulerian Polynomials

Abstract

We study in the present work a recurrence relation, which has long been overlooked, for the $q$-Eulerian polynomial $A_n^{{\rm des},{\rm inv}}(t,q) =\sum_{\sigma\in\mathfrak{S}_n} t^{{\rm des}(\sigma)}q^{{\rm inv}(\sigma)}$, where ${\rm des}(\sigma)$ and ${\rm inv}(\sigma)$ denote, respectively, the descent number and inversion number of $\sigma$ in the symmetric group $\mathfrak{S}_n$ of degree $n$. We give an algebraic proof and a combinatorial proof of the recurrence relation.