Some coefficient sequences related to the descent polynomial

TL;DR

This paper investigates the coefficient sequences of descent polynomials, proving conjectures and identifying zero-free regions, thereby advancing understanding of permutation descent structures.

Contribution

It proves conjectures related to descent polynomial coefficients and describes zero-free regions, offering new insights into permutation descent polynomials.

Findings

Proved conjectures on coefficient sequences

Identified zero-free regions for descent polynomials

Enhanced understanding of descent polynomial properties

Abstract

The descent polynomial of a finite $I\subseteq \mathbb{Z}^+$ is the polynomial $d(I,n)$, for which the evaluation at $n>\max(I)$ is the number of permutations on $n$ elements, such that $I$ is the set of indices where the permutation is descending. In this paper we will prove some conjectures concerning coefficient sequences of $d(I,n)$. As a corollary we will describe some zero-free regions for the descent polynomial.