### 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 language | English (US) |
---|---|

Pages (from-to) | 2921-2966 |

Number of pages | 46 |

Journal | Advances in Mathematics |

Volume | 225 |

Issue number | 6 |

DOIs | |

State | Published - Dec 20 2010 |

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### Keywords

- Permutation statistics
- Symmetric functions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*225*(6), 2921-2966. https://doi.org/10.1016/j.aim.2010.05.009

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 225, no. 6, pp. 2921-2966. https://doi.org/10.1016/j.aim.2010.05.009

}

TY - JOUR

T1 - Eulerian quasisymmetric functions

AU - Shareshian, John

AU - Galloway, Michelle L

PY - 2010/12/20

Y1 - 2010/12/20

N2 - 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.

AB - 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.

KW - Permutation statistics

KW - Symmetric functions

UR - http://www.scopus.com/inward/record.url?scp=77957754347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957754347&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.05.009

DO - 10.1016/j.aim.2010.05.009

M3 - Article

AN - SCOPUS:77957754347

VL - 225

SP - 2921

EP - 2966

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 6

ER -