### Abstract

We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Q_{λ,j} is Schur-positive, and moreover that the sequence Q_{λ,j} as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type (q, p)-Eulerian polynomial Aλmaj,des,exc(q,p,q-1t) is t-unimodal.

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

Pages (from-to) | 135-145 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 119 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2012 |

### Fingerprint

### Keywords

- Cycle type
- Eulerian polynomials
- Permutation statistics
- Symmetric functions
- Unimodality

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory, Series A*,

*119*(1), 135-145. https://doi.org/10.1016/j.jcta.2011.07.004

**Unimodality of Eulerian quasisymmetric functions.** / Henderson, Anthony; Galloway, Michelle L.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 119, no. 1, pp. 135-145. https://doi.org/10.1016/j.jcta.2011.07.004

}

TY - JOUR

T1 - Unimodality of Eulerian quasisymmetric functions

AU - Henderson, Anthony

AU - Galloway, Michelle L

PY - 2012/1

Y1 - 2012/1

N2 - We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Qλ,j is Schur-positive, and moreover that the sequence Qλ,j as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type (q, p)-Eulerian polynomial Aλmaj,des,exc(q,p,q-1t) is t-unimodal.

AB - We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Qλ,j is Schur-positive, and moreover that the sequence Qλ,j as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type (q, p)-Eulerian polynomial Aλmaj,des,exc(q,p,q-1t) is t-unimodal.

KW - Cycle type

KW - Eulerian polynomials

KW - Permutation statistics

KW - Symmetric functions

KW - Unimodality

UR - http://www.scopus.com/inward/record.url?scp=80052092606&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052092606&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2011.07.004

DO - 10.1016/j.jcta.2011.07.004

M3 - Article

AN - SCOPUS:80052092606

VL - 119

SP - 135

EP - 145

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -