Some coefficient sequences related to the descent polynomial

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.