Unimodality of Eulerian quasisymmetric functions

Anthony Henderson and Michelle Wachs

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 A_\lambda^{\maj,\des,\exc}(q,p,q^{-1}t) is t-unimodal.

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