Magic Positivity for the Ehrhart Polynomials of Partial Permutohedra

Abstract

For positive integers \(m,n\), the partial permutohedron $\mathcal{P}(m,n)$ is a lattice polytope constructed as the convex hull of vectors in $\{0, 1, \dots, n\}^m$ that have distinct non-zero entries. We prove that for $n \ge m-1$, the Ehrhart polynomial of $\mathcal{P}(m,n)$ is magic positive except for the single case \((m,n)=(2,1)\). In particular, the Ehrhart polynomial of the parking function polytope (integrally equivalent to $\mathcal{P}(m,m-1)$) is magic positive for $m \ge 3$. For $n<m-1$, we discuss the magic positivity of the Ehrhart polynomial of $\mathcal{P}(m,n)$ for $n=1,2,3$. There exist infinitely many counterexamples with $n<m-1$ showing that the Ehrhart polynomial of $\mathcal{P}(m,n)$ is not magic positive. This partially resolves an open problem proposed by Ferroni and Higashitani.