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Séminaire Lotharingien de Combinatoire, 85B.62 (2021), 12 pp.

# Alex McDonough

# A Combinatorial Mapping for the Higher-Dimensional Matrix-Tree Theorem

**Abstract.**
For a natural class of *r* × *n* integer matrices, we construct a non-convex polytope which periodically tiles **R**^{n}. From this tiling, we provide a family of geometrically meaningful maps from a generalized sandpile group to a set of generalized spanning trees which give multijective proofs for several higher-dimensional matrix-tree theorems. In particular, these multijections can be applied to graphs, regular matroids, cell complexes with a torsion-free spanning forest, and representable arithmetic matroids with a multiplicity one basis. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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