A q-analog of the Stirling–Eulerian Polynomials

期刊论文
发表日期：

2024年9月10日

查看全文

Yao Dong,
Zhicong Lin,
Qiongqiong Pan

摘要

In 1974, Carlitz and Scoville introduced the Stirling–Eulerian polynomial $A_n(x,y|\alpha ,\beta )$ $A_{n} (x, y | α, β)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y|\alpha ,\beta )$ $A_{n} (x, y | α, β)$ , denoted $P_n(u_1,u_2,u_3,u_4|\alpha ,\beta )$ $P_{n} (u_{1}, u_{2}, u_{3}, u_{4} | α, β)$ , 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 $P_n(u_1,u_2,u_3,u_4|\alpha ,\beta )$ $P_{n} (u_{1}, u_{2}, u_{3}, u_{4} | α, β)$ , generalizing the work of Carlitz and Scoville. Ji’s formula has many nice consequences, one of which is an intriguing $\gamma $ $γ$ -positivity expansion for $A_n(x,y|\alpha ,\beta )$ $A_{n} (x, y | α, β)$ . 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 $\gamma $ $γ$ -positivity expansion of $A_n(x,y|\alpha ,\beta )$ $A_{n} (x, y | α, β)$ .