Séminaire Lotharingien de Combinatoire, 93B.41 (2025), 12 pp.
Chao Xu and Jiang Zeng
Variations of the (α,t)-Eulerian Polynomials and Gamma-Positivity
Abstract.
We define a multivariable generalization of the Eulerian polynomials using linear and descent based statistics of permutations and establish the connection with the
(α,t)-Eulerian polynomials based on cyclic and excedance based statistics of permutations.
As applications of this connection,
we obtain the exponential generating function for the multivariable Eulerian polynomials and
γ-positive formulas of two variants of Eulerian polynomials.
We also show that enumerating the cycle André permutations with respect to the number of drops, fixed points and cycles gives rise to the normalised
γ-vectors of the (α,t)-Eulerian polynomials.
Our result generalizes and unifies several recent results in the literature.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
The following versions are available: