A $q$-analog of the Stirling-Eulerian Polynomials

Abstract

In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y|α,β)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y|α,β)$, denoted $P_n(u_1,u_2,u_3,u_4|α,β)$, which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. Using Chen's context-free grammar calculus, Ji proved a formula for the generating function of $P_n(u_1,u_2,u_3,u_4|α,β)$, generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing $γ$-positivity expansion for $A_n(x,y|α,β)$. In this paper, we prove a $q$-analog of Ji's formula by using Gessel's $q$-compositional formula and provide a combinatorial approach to her $γ$-positivity expansion of $A_n(x,y|α,β)$.