Eulerian quasisymmetric functions

John Shareshian, Michelle L Galloway

Research output: Contribution to journalArticle

27 Citations (Scopus)

Abstract

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize 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 including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.

Original languageEnglish (US)
Pages (from-to)2921-2966
Number of pages46
JournalAdvances in Mathematics
Volume225
Issue number6
DOIs
StatePublished - Dec 20 2010

Fingerprint

Quasi-symmetric Functions
Major Index
Q-analogue
Joint Distribution
Symmetric Functions
Descent
Euler
Permutation
Permutation Statistics
Exponential Generating Function
Analogue
Cycle
Unimodality
Polynomial
Toric Varieties
Multiset
Rearrangement
Poset
Generating Function
Statistic

Keywords

  • Permutation statistics
  • Symmetric functions

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Eulerian quasisymmetric functions. / Shareshian, John; Galloway, Michelle L.

In: Advances in Mathematics, Vol. 225, No. 6, 20.12.2010, p. 2921-2966.

Research output: Contribution to journalArticle

Shareshian, John ; Galloway, Michelle L. / Eulerian quasisymmetric functions. In: Advances in Mathematics. 2010 ; Vol. 225, No. 6. pp. 2921-2966.
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