Gamma positivity of variations of $(\alpha,t)$-Eulerian polynomials

Abstract

In 1977 Carlitz and Scoville introduced the cycle $(\alpha,t)$-Eulerian polynomials $A^{\mathrm{cyc}}_n(x,y, t\,|\,\alpha)$ by enumerating permutations with respect to the number of excedances, drops, fixed points and cycles. In this paper, we introduce a nine-variable generalization of the Eulerian polynomials $A_n(u_1,u_2,u_3,u_4, f, g, t\,|\,\alpha, \beta)$ in terms of descent based statistics of permutations and prove a connection formula between these two kinds of generalized Eulerian polynomials. By exploring the connection formula, we derive plainly the exponential generating function of the latter polynomials and various $\gamma$-positive formulas for variants of Eulerian polynomials. In particular, our results unify and strengthen the recent results by Ji and Ji-Lin. In related work to the transition matrix between the Specht and web bases, Hwang, Jang and Oh recently introduced the web permutations, which can be characterised by cycle Andr\'e permutations. We show that enumerating the latter permutations with respect to the number of drops, fixed points and cycles gives rise to the normalised $\gamma$-vectors of the $(\alpha,t)$-Eulerian polynomials. Our result generalizes and unifies several known results in the literature.