Generalized snake posets, order polytopes, and lattice-point enumeration
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915820662161408 |
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| author | Lee, Eon Vindas-Meléndez, Andrés R. Wang, Zhi |
| author_facet | Lee, Eon Vindas-Meléndez, Andrés R. Wang, Zhi |
| contents | Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the Ehrhart polynomials of order polytopes of generalized snake posets along with a computation of their Gorenstein index. Then we give a combinatorial description of the chain polynomial of generalized snake posets as a direction to obtain the $h^*$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^*$-polynomial of the order polytopes of the two extremal examples of generalized snake posets, namely the ladder and regular snake poset. We then provide a recursive formula for the $h^*$-polynomial of any generalized snake posets and show that the $h^*$-vectors are entry-wise bounded by the $h^*$-vectors of the two extremal cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18695 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized snake posets, order polytopes, and lattice-point enumeration Lee, Eon Vindas-Meléndez, Andrés R. Wang, Zhi Combinatorics 05A10, 05A15, 05A19, 52B05, 52B11, 52B15, 52B20 Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the Ehrhart polynomials of order polytopes of generalized snake posets along with a computation of their Gorenstein index. Then we give a combinatorial description of the chain polynomial of generalized snake posets as a direction to obtain the $h^*$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^*$-polynomial of the order polytopes of the two extremal examples of generalized snake posets, namely the ladder and regular snake poset. We then provide a recursive formula for the $h^*$-polynomial of any generalized snake posets and show that the $h^*$-vectors are entry-wise bounded by the $h^*$-vectors of the two extremal cases. |
| title | Generalized snake posets, order polytopes, and lattice-point enumeration |
| topic | Combinatorics 05A10, 05A15, 05A19, 52B05, 52B11, 52B15, 52B20 |
| url | https://arxiv.org/abs/2411.18695 |