Unimodality of Eulerian quasisymmetric functions

TL;DR

This paper proves two conjectures related to Eulerian quasisymmetric functions, establishing their Schur-positivity, Schur-unimodality, and the t-unimodality of associated polynomials, advancing understanding of their algebraic and combinatorial properties.

Contribution

The paper proves two conjectures about Eulerian quasisymmetric functions, demonstrating their Schur-positivity, Schur-unimodality, and t-unimodality of related polynomials, which were previously unconfirmed.

Findings

Proved Schur-positivity of cycle type Eulerian quasisymmetric functions.

Established Schur-unimodality of the sequence of these functions.

Confirmed t-unimodality of the cycle type (q,p)-Eulerian polynomial.

Abstract

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