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arXiv · 1604.04140

Multivariate P-Eulerian polynomials

Petter Brändén·Madeleine Leander
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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.

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BibTeXRIS

Petter Brändén, Madeleine Leander. 2016-04-14. Multivariate P-Eulerian polynomials. https://arxiv.org/abs/1604.04140

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