For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is a certain integral polytope in $\mathbb{R}^m$, which can be defined as the convex hull of the vectors from $\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. For $n=m-1$, $\mathcal{P}(m,m-1)$ is (after translation by $(1,\ldots,1)$) the polytope $P_m$ of parking functions of length $m$, and for $n\ge m$, $\mathcal{P}(m,n)$ is combinatorially equivalent to an $m$-stellohedron. The main result of this paper is an explicit expression for the Ehrhart polynomial of $\mathcal{P}(m,n)$ for any $m$ and $n$ with $n\ge m-1$. The result confirms the validity of a conjecture for this Ehrhart polynomial in arXiv:2207.14253, and the $n=m-1$ case also answers a question of Stanley regarding the number of integer points in $P_m$. The proof of the result involves transforming $\mathcal{P}(m,n)$ to a unimodularly equivalent polytope in $\mathbb{R}^{m+1}$, obtaining a decomposition of this lifted version of $\mathcal{P}(m,n)$ with $n\ge m-1$ as a Minkowski sum of dilated coordinate simplices, applying a result of Postnikov for the number of integer points in generalized permutohedra of this form, observing that this gives an expression for the Ehrhart polynomial of $\mathcal{P}(m,n)$ with $n\ge m-1$ as an edge-weighted sum over graphs (with loops and multiple edges permitted) on $m$ labelled vertices in which each connected component contains at most one cycle, and then applying standard techniques for the enumeration of such graphs.
在这篇文章中，我们反驳了De Loera，Haws and Köppe（2007）关于拟阵多面体的Ehrhart多项式系数的正性以及Castillo and Liu（2015）关于广义置换面体的Ehrhart多项式系数的正性的断言。我们建设性地证明了对于每一个$n\geq 19$都存在$n$元素上的非Ehrhart正的连通拟阵.证明了对每个$k\geq 3$，存在秩为$k$的连通拟阵，且它们不是Ehrhart正的.我们的证明依赖于我们以前的结果的几何解释的操作电路超平面松弛和我们的公式的Ehrhart多项式的超单形和极小拟阵。这使我们能够给出一个精确的表达式的Ehrhart多项式的所有稀疏铺路拟阵，一类拟阵，这是被证明是占主导地位的，其中包含的反例所产生的我们的建设。
School of Mathematics, Cardiff University（卡迪夫大学数学学院）；Departamento de Matemáticas, Pontificia Universidad Católica de Chile（智利天主教大学数学系）；Department of Mathematics and Computer Science, Saint Mary’s College of California（加州圣玛丽学院数学与计算机科学系）；Department of Mathematics and Statistics, Villanova University（维拉诺瓦大学数学与统计学系）；Department of Mathematics, University of California Santa Cruz（加州大学圣克鲁兹分校数学系）；Department of Mathematical Sciences, University of Wisconsin Milwaukee（威斯康星大学密尔沃基分校数学科学系）；Department of Mathematics, Central College（中央学院数学系）
