Unimodality of Eulerian quasisymmetric functions

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.