Eulerian quasisymmetric functions and poset topology

TL;DR

This paper introduces Eulerian quasisymmetric functions that generalize classical Eulerian polynomials, providing new formulas and $q$-analogs for permutation statistics and connecting to poset topology and representation theory.

Contribution

It defines Eulerian quasisymmetric functions, derives a generating function formula, and establishes new $q$-analogs and symmetry properties, extending classical results in permutation enumeration.

Findings

Derived a $q$-analog of the exponential Eulerian polynomial formula.

Computed the joint distribution of excedance number and major index.

Connected Eulerian quasisymmetric functions to poset homology and representation theory.

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 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 and in Stanley's work on symmetric chromatic polynomials. Here we present yet another occurence in connection with the homology of a poset introduced by Bj\"orner and Welker.