Hypergraphic zonotopes and acyclohedra
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917969200676864 |
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| author | Pohoata, Cosmin Zhu, Daniel G. |
| author_facet | Pohoata, Cosmin Zhu, Daniel G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypergraphic zonotopes and acyclohedra Pohoata, Cosmin Zhu, Daniel G. Combinatorics 52B05 (Primary) 05C20, 05E45 (Secondary) 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. |
| title | Hypergraphic zonotopes and acyclohedra |
| topic | Combinatorics 52B05 (Primary) 05C20, 05E45 (Secondary) |
| url | https://arxiv.org/abs/2503.21752 |