ar X iv : 1 60 4 . 04 14 0 v 1 [ m at h . C O ] 1 4 A pr 2 01 6 MULTIVARIATE P - EULERIAN POLYNOMIALS

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.