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 normalized volume of \({{\mathcal {P}}}(m,4)\) with arbitrary m, and the Ehrhart polynomial of \({{\mathcal {P}}}(m,n)\) with fixed \(m\le 4\) and arbitrary \(n\ge m-1\) .
