A q-analog of the Stirling–Eulerian Polynomials.

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) . [ABSTRACT FROM AUTHOR]