I would like to draw your attention to the short beautiful paper Hypergraphic Zonotopes and Acyclohedra by Cosmin Pohoata and Daniel G. Zhu. The paper introduces higher-uniformity analogue of graphic zonotopes and permutohedra, and provides formulas for their volume, and, more generally, for their Ehrhart polynomials. These formulas are related to the notions of hypertrees and their enumeration that I studied in the early 80s and were discussed in some of the first posts in the blog (I, II, III, IV,V). Cosmin and Daniel also connect the hypergraphic zonotopes to high-dimensional analogs of acyclic orientations of graphs introduced by Linial and Morgenstern.
The graphic zonotopes and permutahedron
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be the standard basis for Image may be NSFW.
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. Given a graph Image may be NSFW.
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with Image may be NSFW.
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vertices Image may be NSFW.
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, the graphic zonotope of Image may be NSFW.
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is adjacent to Image may be NSFW.
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. If Image may be NSFW.
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-dimensional polytope, and remarkably its volume is the number of spanning trees of Image may be NSFW.
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permutations of the vector Image may be NSFW.
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.
Hypergraphs and Hypertrees
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, where the Image may be NSFW.
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form a basis to an Image may be NSFW.
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dimensional space, and the sum is over all edges Image may be NSFW.
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of Image may be NSFW.
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). This zonotope is called the hypergraphic zonotope of Image may be NSFW.
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acyclic spanning subcomplexes with a complete 1-skeleton). The weight for a hypertree Image may be NSFW.
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is the size of the torsion group Image may be NSFW.
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. (These formulas extend to Image may be NSFW.
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-uniform hypergraphs for every Image may be NSFW.
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.)
Hypertournaments
The vertices of the acyclohedron are in 1-1 correspondence to the high-dimensional analogs of tournaments introduced in 2013 by Nati Linial and Avraham Morgenstern. We consider orientations of the faces of the complete Image may be NSFW.
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. Here Image may be NSFW.
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refers to the set of nonnegative real numbers.
Remark
The volume of the acyclohedron corresponding to (d+1)-uniform hypergraphs on Image may be NSFW.
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,
where the sum is over all Image may be NSFW.
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-dimensional simplicial complexes Image may be NSFW.
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on Image may be NSFW.
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-acyclic, namely all their (reduced) homology groups with rational coefficients vanish. It will be interesting to give good estimates for this expression. .
As noted by Cosmin and Daniel, these weights are closely related but not identical to those appearing in an old result of mine. I proved in the early 80s an extension of Cayley’s theorem that refers to the sum of squares of the sizes of the torsion groups (again over the same subcomplexes):
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Daniel G. Zhu, Cosmin Pohoata, me, and Arthur Cayley.