Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915312078684160 |
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| author | Deligeorgaki, Danai McGinnis, Daniel Vindas-Meléndez, Andrés R. |
| author_facet | Deligeorgaki, Danai McGinnis, Daniel Vindas-Meléndez, Andrés R. |
| contents | Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. These results follow from purely enumerative statements, the main one being conjectured by Hanely et. al concerning the enumeration of a certain class of ordered chain forests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01640 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests Deligeorgaki, Danai McGinnis, Daniel Vindas-Meléndez, Andrés R. Combinatorics 52B20, 52B40, 05A19 Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. These results follow from purely enumerative statements, the main one being conjectured by Hanely et. al concerning the enumeration of a certain class of ordered chain forests. |
| title | Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests |
| topic | Combinatorics 52B20, 52B40, 05A19 |
| url | https://arxiv.org/abs/2311.01640 |