Multivariate stable Eulerian polynomials on segmented permutations

Abstract

Recently, Nunge studied Eulerian polynomials on segmented permutations, namely 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 Brändén’s stable multivariate Eulerian polynomials. By acting on Brändén’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.