Unimodality of Eulerian quasisymmetric functions

Anthony Henderson; Michelle L. Wachs

doi:10.1016/j.jcta.2011.07.004

Unimodality of Eulerian quasisymmetric functions

Anthony Henderson, Michelle L. Wachs

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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	https://doi.org/10.1016/j.jcta.2011.07.004
State	Published - Jan 1 2012

Keywords

Cycle type
Eulerian polynomials
Permutation statistics
Symmetric functions
Unimodality

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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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 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.",

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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

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