Unimodality of Eulerian quasisymmetric functions

Details Unimodality of Eulerian quasisymmetric functions Unimodality of Eulerian quasisymmetric functions

2011-01-24

arxiv.org/abs/1101.4441v1

J. Combin. Theory Ser. A 119 (2012), no. 1, 135-145

Mathematics

Combinatorics

15 pages

Scientific paper

10.1016/j.jcta.2011.07.004

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.

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