2025/03/27 by Pohoata, Cosmin, Zhu, Daniel G.
#05E45 (Secondary) #52B05 (Primary) 05C20 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.21752
We introduce a higher-uniformity analogue of graphic zonotopes and permutohedra. Specifically, given a (d+1)-uniform hypergraph H, we define its hypergraphic zonotope ZH, and when H is the complete (d+1)-uniform hypergraph K(d+1)n, we call its hypergraphic zonotope the acyclohedron An,d. We express the volume of ZH 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.