A $q$-analog of the Stirling-Eulerian Polynomials
A $q$-analog of the Stirling-Eulerian Polynomials
In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y|\alpha,\beta)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y|\alpha,\beta)$, denoted $P_n(u_1,u_2,u_3,u_4|\alpha,\beta)$, which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. …