Multivariate Stable Eulerian Polynomials on Segmented Permutations

TL;DR

This paper proves the stability of generalized Eulerian polynomials on segmented permutations, confirming a conjecture about their unimodal coefficient sequences using advanced multivariate polynomial techniques.

Contribution

It introduces a stability proof for generalized Eulerian polynomials and develops a new approach to derive Sturm sequences from stable polynomials.

Findings

Proved the stability of generalized Eulerian polynomials.

Confirmed Nunge's conjecture on unimodality.

Developed a method to obtain Sturm sequences from stable polynomials.

Abstract

Recently, Nunge studied Eulerian polynomials on segmented permutations, namely \emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunge's conjecture. Our proof is based on Br\"and\'en's stable multivariate Eulerian polynomials. By acting on Br\"and\'en's polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunge's conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.