Variations of the $α$-Eulerian polynomials and gamma positivity

Chao Xu, Jiang Zeng

Published 2024-04-12Version 1

We prove a connection formula between two multivariate generalizations of the Eulerian polynomials $A^{\cyc}_n(x,y, t\,|\,\alpha)$ and $A_n(u_1,u_2,u_3,u_4, f, g, t\,|\,\alpha, \beta)$, which enumerate permutations related to excedance and descent based statistics respectively. By exploring this connection, we derive the exponential generating function of the latter polynomials and several $\gamma$-positivity formulas for variants of Eulerian polynomials. In particular, our results generalise the main results in two recent papers by Ji \cite{Ji23} and Ji-Lin \cite{JL23}. Our proofs are combinatorial in nature and involve Foata's fundamental transformation and a cyclic analogue of valley-hopping.