Partial permutohedra

Abstract

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker [arXiv:2012.09901]. For positive integers m and n, the partial permutohedron P(m, n) is the convex hull of all vectors in {0, 1, . . . , n} m with distinct nonzero entries. We study the face lattice, volume and Ehrhart polynomial of 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 P(m, n) and certain chains of subsets of {1, . . . , 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 P(m, n). For any m and n with n ≥ m − 1, we use a pyramidal subdivision of P(m, n) to obtain a recursive formula for the volume of P(m, n). We also use a sculpting process (in which P(m, n) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of P(m, n) with arbitrary m and fixed n ≤ 3, the volume of P(m, 4) with arbitrary m, and the Ehrhart polynomial of P(m, n) with fixed m ≤ 4 and arbitrary n ≥ m − 1.