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In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial Aₙ (x, y|, ) 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|, ), denoted Pₙ (u₁, u₂, u₃, u₄|, ), 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|, ), generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing -positivity expansion for Aₙ (x, y|, ). 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 -positivity expansion of Aₙ (x, y|, ).
Dong et al. (Mon,) studied this question.