Eulerian polynomials and descent statistics

TL;DR

This paper establishes identities connecting permutation descent statistics with Eulerian polynomials, introduces q-analogues with inversion and major index tracking, and explores their relations to type B Eulerian and flag descent polynomials using advanced combinatorial methods.

Contribution

It extends known identities by incorporating q-analogues and signed permutations, employing noncommutative symmetric functions and group actions for new combinatorial insights.

Findings

Derived identities linking descent polynomials to Eulerian polynomials

Developed q-exponential generating functions with inversion and major index

Connected descent statistics to type B Eulerian and flag descent polynomials

Abstract

We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Br\"and\'en. Additionally, we find $q$-exponential generating functions for $q$-analogues of these descent statistic polynomials that also keep track of the inversion number or inverse major index. We also present identities relating several of these descent statistic polynomials to refinements of type B Eulerian polynomials and flag descent polynomials by the number of negative letters of a signed permutation. Our methods include permutation enumeration techniques involving noncommutative symmetric functions, Br\"and\'en's modified Foata-Strehl action, and a group action of Petersen on signed permutations. Notably, the modified Foata-Strehl action yields an analogous relation between Narayana polynomials and the joint distribution of the peak number and descent number over 231-avoiding permutations, which we also interpret in terms of binary trees and Dyck paths.