A q-analog of the Stirling-Eulerian Polynomials

In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial Aₙ(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 Aₙ(x,y|α,β), denoted Pₙ(u₁,u₂,u₃,u₄|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 Pₙ(u₁,u₂,u₃,u₄|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 Aₙ(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 Aₙ(x,y|α,β).