Descent polynomials

TL;DR

This paper investigates the properties of descent polynomials, which count permutations with a given descent set, exploring their coefficients, roots, and parallels with peak polynomials, and proposing future research directions.

Contribution

It provides new insights into the properties of descent polynomials, including their coefficients and roots, and draws parallels with peak polynomials, filling gaps in the literature.

Findings

Descent polynomials have interesting coefficient and root structures.

Parallel properties between descent and peak polynomials are identified.

Open conjectures and questions for further research are proposed.

Abstract

Let $n$ be a nonnegative integer and $I$ be a finite set of positive integers. In 1915, MacMahon proved that the number of permutations in the symmetric group $\mathfrak{S}_n$ with descent set $I$ is a polynomial in $n$. We call this the descent polynomial. However, basic properties of these polynomials such as a description of their coefficients and roots do not seem to have been studied in the literature. Much more recently, in 2013, Billey, Burdzy, and Sagan showed that the number of elements of $\mathfrak{S}_n$ with peak set $I$ is a polynomial in $n$ times a certain power of two. Since then, there have been a flurry of papers investigating properties of this peak polynomial. The purpose of the present paper is to study the descent polynomial. We will see that it displays some interesting parallels with its peak relative. Conjectures and questions for future research are scattered throughout.