Hypergraphic zonotopes and acyclohedra
Cosmin Pohoata, Daniel G. Zhu
TL;DR
This work extends graphic zonotopes and permutohedra to higher uniformity by introducing the hypergraphic zonotope $\mathcal{Z}_H$ for a $(d+1)$-uniform hypergraph and the acyclohedron $\mathcal{A}_{n,d}$ for the complete case. The authors derive an Ehrhart-theoretic formula: the lattice-point count $L(\mathcal{Z}_H,t)$ is the sum over spanning hyperforests $F$ of $H$ of the torsion $|\tilde{H}_{d-1}(F;\mathbb{Z})_{\mathrm{tors}}|$ times $t^{|E(F)|}$, which implies that the volume of $\mathcal{Z}_H$ equals $\sum_T |\tilde{H}_{d-1}(T;\mathbb{Z})|$ over spanning hypertrees $T$—a Kalai-type but with non-squared homology weights. For the complete hypergraph, the vertices of $\mathcal{A}_{n,d}$ correspond to $d$-dimensional acyclic hypertournaments, tying hypertree counts to Linial–Morganstern’s acyclic orientations. The paper also provides a sign-pattern framework that identifies faces and vertices of $\mathcal{Z}_H$, and discusses dualities and facets of acyclohedra, highlighting rich combinatorial/topological structure in high-uniformity settings. Together, these results offer a polytopal perspective on high-dimensional hypergraph invariants and generalized Cayley-type formulas with potential combinatorial and geometric applications.
Abstract
We introduce a higher-uniformity analogue of graphic zonotopes and permutohedra. Specifically, given a $(d+1)$-uniform hypergraph $H$, we define its hypergraphic zonotope $\mathcal{Z}_H$, and when $H$ is the complete $(d+1)$-uniform hypergraph $K^{(d+1)}_n$, we call its hypergraphic zonotope the acyclohedron $\mathcal{A}_{n,d}$. We express the volume of $\mathcal{Z}_H$ as a homologically weighted count of the spanning $d$-dimensional hypertrees of $H$, which is closely related to Kalai's generalization of Cayley's theorem in the case when $H=K^{(d+1)}_n$ (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 Morganstern for complete hypergraphs.