Stable multivariate $W$-Eulerian polynomials

TL;DR

This paper proves a multivariate extension of real-rootedness for Eulerian polynomials of type B, introduces stability-preserving recurrences, and extends results to colored permutations and affine types, while posing open problems.

Contribution

It provides a multivariate strengthening of real-rootedness results for Eulerian polynomials using stability theory and extends these to new permutation types and affine cases.

Findings

Multivariate Eulerian polynomials satisfy stability-preserving recurrences.

Extended real-rootedness results to colored permutations and affine types A and C.

Identified open problems and methods for future research on types D, B, and D affine Eulerian polynomials.

Abstract

We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type $B$ is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability-a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types $A$ and $C$. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type $D$, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types $B$ and $D$, we indicate some methods of attack and pose some related open problems.