# ALGEBRAIC COMBINATORICS

A family of matrix-tree multijections
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 795-822.

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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.181
Classification: 05B45,  52C40,  52C22,  05E45
Keywords: sandpile group, multijection, arithmetic matroid
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McDonough, Alex. A family of matrix-tree multijections. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 795-822. doi : 10.5802/alco.181. https://alco.centre-mersenne.org/articles/10.5802/alco.181/

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