  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.
