Multivariate P-Eulerian polynomials
Abstract
The P-Eulerian polynomial counts the linear extensions of a labeled partially ordered set, P, by their number of descents. It is known that the P-Eulerian polynomials are real-rooted for various classes of posets P. The purpose of this paper is to extend these results to polynomials in several variables. To this end we study multivariate extensions of P-Eulerian polynomials and prove that for certain posets these polynomials are stable, i.e., non-vanishing whenever all variables are in the upper half-plane of the complex plane. A natural setting for our proofs is the Malvenuto-Reutenauer algebra of permutations (or the algebra of free quasi-symmetric functions). In the process we identify an algebra on Dyck paths, which to our knowledge has not been studied before.
Cite
@article{arxiv.1604.04140,
title = {Multivariate P-Eulerian polynomials},
author = {Petter Brändén and Madeleine Leander},
journal= {arXiv preprint arXiv:1604.04140},
year = {2016}
}