On two unimodal descent polynomials

TL;DR

This paper proves the unimodality of descent polynomials for separable permutations and derangements, explores their gamma-coefficients and spiral property, and conjectures their real-rootedness.

Contribution

It establishes unimodality for these descent polynomials, analyzes their gamma-coefficients, and introduces the spiral property, proposing real-rootedness conjectures.

Findings

Descent polynomials of separable permutations are unimodal.

Descent polynomials of derangements are unimodal and spiral.

Gamma-coefficients of separable permutation polynomials are positive.

Abstract

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.