The descent polynomial of a finite $I\subseteq {\mathbb{Z}}^{+}$ is the polynomial $d(I,n)$, for which the evaluation at $n>max\left(I\right)$ 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.

中文翻译：

---

一些与下降多项式相关的系数序列

有限元的下降多项式 $\mathrm{一世}\subseteq {\mathbb{Z}}^{+}$ 是多项式 $d(\mathrm{一世},n)$，其中评估在 $n>最大限度\left(\mathrm{一世}\right)$ 是排列的数量 $n$ 元素，使得 $\mathrm{一世}$是排列下降的索引集。在本文中，我们将证明一些关于系数序列的猜想$d(\mathrm{一世},n)$. 作为推论，我们将描述下降多项式的一些无零区域。