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arXiv · 2503.21752

Hypergraphic zonotopes and acyclohedra

Cosmin Pohoata·Daniel G. Zhu
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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.

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Cosmin Pohoata, Daniel G. Zhu. 2025-03-27. Hypergraphic zonotopes and acyclohedra. https://arxiv.org/abs/2503.21752

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