  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.
