Partial permutohedra

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers \(m\) and \(n\), the partial permutohedron \(\mathcal{P}(m,n)\) is the convex hull of all vectors in \(\{0,1,\ldots,n\}^m\) whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of \(\mathcal{P}(m,n)\), and our methods and results include the following. For any \(m\) and \(n\), we obtain a bijection between the nonempty faces of \(\mathcal{P}(m,n)\) and certain chains of subsets of \(\{1,\dots,m\}\), thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the \(h\)-polynomial of \(\mathcal{P}(m,n)\). For any \(m\) and \(n\) with \(n\ge m-1\), we use a pyramidal subdivision of \(\mathcal{P}(m,n)\) to establish a recursive formula for the normalized volume of \(\mathcal{P}(m,n)\), from which we then obtain closed expressions for this volume. We also use a sculpting process (in which \(\mathcal{P}(m,n)\) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of \(\mathcal{P}(m,n)\) with arbitrary \(m\) and fixed \(n\le 3\), the volume of \(\mathcal{P}(m,4)\) with arbitrary \(m\), and the Ehrhart polynomial of \(\mathcal{P}(m,n)\) with fixed \(m\le4\) and arbitrary \(n\ge m-1\).