## Eulerian quasisymmetric functions

**
John Shareshian and Michelle L. Wachs**
**Abstract:**
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric
functions}, which have the property of specializing to enumerators for the joint distribution of
the permutation statistics, major index and excedance number on permutations of fixed cycle type.
This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used
to study the joint distribution of major index and descent number on permutations of fixed cycle
type. Our central result is a formula for the generating function for the Eulerian quasisymmetric
functions, which specializes to a new and surprising $q$-analog of a classical formula of Euler
for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the
joint distribution of excedance number and major index, the only of the four important
Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian
quasisymmetric functions also yields results that include the descent statistic and refine
results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and
quasisymmetric function analogs of classical results on the symmetry and unimodality of the
Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have
occurred in various representation theoretic and enumerative contexts such as in MacMahon's study
of multiset derangements, in work of Procesi and Stanley on toric varieties of Coxeter
complexes, in Stanley's work on symmetric chromatic polynomials, and in the work of the authors
on the homology of a certain poset introduced by Bj\"orner and Welker.

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