Eulerian polynomials and descent statistics

Abstract

We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Brändén. 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, the 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.