Eulerian polynomials and descent statistics

Yan Zhuang

Published 2016-10-23Version 1

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 then show that several of these descent statistic polynomials are closely related 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.