Hypergraphic zonotopes and acyclohedra

Abstract We introduce a higher uniformity analogue of graphic zonotopes and permutohedra. Specifically, given a ‐uniform hypergraph , we define its hypergraphic zonotope , and when is the complete ‐uniform hypergraph , we call its hypergraphic zonotope the acyclohedron . We express the volume of as a homologically weighted count of the spanning ‐dimensional hypertrees of , which is closely related to Kalai's generalization of Cayley's theorem in the case when (but which, curiously, is not the same). We also relate the vertices of hypergraphic zonotopes to a notion of acyclic orientations previously studied by Linial and Morgenstern for complete hypergraphs.