### Abstract

We derive a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This arose in our work on poset topology and was presented as a conjecture at the second author's FPSAC 2006 lecture. We have since proved our conjecture and a symmetric function generalization of it. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. Our proof involves an intriguing new class of symmetric functions and a bijection of Gessel and Reutenauer, which can be viewed as a necklace analog of Stanley's theory of P-partitions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker, (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts, (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville and Vaughan and by Dollhopf, Goulden and Greene, and (4) Stanley's chromatic symmetric functions.

Original language | English (US) |
---|---|

Title of host publication | FPSAC'07 - 19th International Conference on Formal Power Series and Algebraic Combinatorics |

State | Published - 2007 |

Event | 19th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'07 - Tianjin, China Duration: Jul 2 2007 → Jul 6 2007 |

### Other

Other | 19th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'07 |
---|---|

Country | China |

City | Tianjin |

Period | 7/2/07 → 7/6/07 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*FPSAC'07 - 19th International Conference on Formal Power Series and Algebraic Combinatorics*

**Q-Eulerian polynomials : Excedance number and major index.** / Shareshian, John; Galloway, Michelle L.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*FPSAC'07 - 19th International Conference on Formal Power Series and Algebraic Combinatorics.*19th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'07, Tianjin, China, 7/2/07.

}

TY - GEN

T1 - Q-Eulerian polynomials

T2 - Excedance number and major index

AU - Shareshian, John

AU - Galloway, Michelle L

PY - 2007

Y1 - 2007

N2 - We derive a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This arose in our work on poset topology and was presented as a conjecture at the second author's FPSAC 2006 lecture. We have since proved our conjecture and a symmetric function generalization of it. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. Our proof involves an intriguing new class of symmetric functions and a bijection of Gessel and Reutenauer, which can be viewed as a necklace analog of Stanley's theory of P-partitions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker, (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts, (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville and Vaughan and by Dollhopf, Goulden and Greene, and (4) Stanley's chromatic symmetric functions.

AB - We derive a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This arose in our work on poset topology and was presented as a conjecture at the second author's FPSAC 2006 lecture. We have since proved our conjecture and a symmetric function generalization of it. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. Our proof involves an intriguing new class of symmetric functions and a bijection of Gessel and Reutenauer, which can be viewed as a necklace analog of Stanley's theory of P-partitions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker, (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts, (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville and Vaughan and by Dollhopf, Goulden and Greene, and (4) Stanley's chromatic symmetric functions.

UR - http://www.scopus.com/inward/record.url?scp=84860756049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860756049&partnerID=8YFLogxK

M3 - Conference contribution

BT - FPSAC'07 - 19th International Conference on Formal Power Series and Algebraic Combinatorics

ER -