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

In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y\mathrm{\mid }\alpha ,\beta )$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y\mathrm{\mid }\alpha ,\beta )$, denoted $P_n(u_1,u_2,u_3,u_4\mathrm{\mid }\alpha ,\beta )$, 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 $P_n(u_1,u_2,u_3,u_4\mathrm{\mid }\alpha ,\beta )$, generalizing the work of Carlitz and Scoville. Ji’s formula has many nice consequences, one of which is an intriguing $\gamma$-positivity expansion for $A_n(x,y\mathrm{\mid }\alpha ,\beta )$. 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 $\gamma$-positivity expansion of $A_n(x,y\mathrm{\mid }\alpha ,\beta )$.