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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.15253 |
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| _version_ | 1866909014263070720 |
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| author | Beck, Matthias Klivans, Caroline Ross, Dustin |
| author_facet | Beck, Matthias Klivans, Caroline Ross, Dustin |
| contents | Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron. We study a modification of the rational functions associated to the vertices of P depending on a given matroid M. Upon summing these rational functions, we describe how the resulting Laurent polynomial Q_M(P) behaves in certain ways like the lattice points of P, exhibiting natural recursive and reciprocity behaviors. Furthermore, upon evaluating Q_M(P) at 1, we recover the matroid Euler characteristic of Larson, Li, Payne, and Proudfoot, so the combinatorial approach in this paper gives new insight into studying these quantities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A matroidal twist on a formula of Brion Beck, Matthias Klivans, Caroline Ross, Dustin Combinatorics Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron. We study a modification of the rational functions associated to the vertices of P depending on a given matroid M. Upon summing these rational functions, we describe how the resulting Laurent polynomial Q_M(P) behaves in certain ways like the lattice points of P, exhibiting natural recursive and reciprocity behaviors. Furthermore, upon evaluating Q_M(P) at 1, we recover the matroid Euler characteristic of Larson, Li, Payne, and Proudfoot, so the combinatorial approach in this paper gives new insight into studying these quantities. |
| title | A matroidal twist on a formula of Brion |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.15253 |