On the Real-rootedness of the Descent Polynomials of $(n-2)$-Stack Sortable Permutations

TL;DR

This paper provides an alternative proof that the descent polynomials of (n-2)-stack sortable permutations have only real zeros, using the theory of s-Eulerian polynomials.

Contribution

It offers a new proof of Brndenf3n's result leveraging the recent theory of s-Eulerian polynomials.

Findings

Confirmed the real-rootedness of descent polynomials for (n-2)-stack sortable permutations.

Connected the problem to the theory of s-Eulerian polynomials.

Provided an alternative proof to a known conjecture.

Abstract

B\'ona conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Br\"and\'en proved this conjecture by establishing a more general result. In this paper, we give another proof of Br\"and\'en's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.