Eulerian polynomials, Stirling permutations and increasing trees

TL;DR

This paper explores advanced generalizations of Eulerian polynomials, providing new expansion formulas and combinatorial interpretations related to permutations and increasing trees, enhancing understanding of their statistical distributions.

Contribution

It introduces new expansion formulas for multivariate Eulerian polynomials and establishes e-positivity results with combinatorial interpretations involving increasing trees.

Findings

Derived an expansion formula for trivariate Eulerian polynomials.

Proved e-positivity of multivariate k-th order Eulerian polynomials.

Provided combinatorial interpretations for expansion coefficients.

Abstract

We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for the joint distribution of descents, big ascents and successions of permutations. And then, inspired by the work of Chen and Fu on the trivariate second-order Eulerian polynomials, we show the e-positivity of the multivariate k-th order Eulerian polynomials, which are the enumerators for the joint distribution of ascents, descents and j-plateaux of k-Stirling permutations. We provide combinatorial interpretations for the coefficients of these two expansions in terms of increasing trees.