  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.
