Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911578895417344 |
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| author | Liu, Feihu Xin, Guoce Zhang, Zihao |
| author_facet | Liu, Feihu Xin, Guoce Zhang, Zihao |
| contents | The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d\geq 14$, Liu and Tsuchiya proved that there exists a non-Ehrhart positive order polytope of dimension $d$. They also proved that any order polytope of dimension $d\leq 11$ is Ehrhart positive. We confirm that any order polytope of dimension $12$ or $13$ is Ehrhart positive. This solves an open problem proposed by Liu and Tsuchiya. Besides, we also verify that any $h^{*}$-polynomial of order polytope of dimension $d\leq 13$ is real-rooted. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_07164 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive Liu, Feihu Xin, Guoce Zhang, Zihao Combinatorics Primary 05A15, Secondary 06A07, 68R05 The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d\geq 14$, Liu and Tsuchiya proved that there exists a non-Ehrhart positive order polytope of dimension $d$. They also proved that any order polytope of dimension $d\leq 11$ is Ehrhart positive. We confirm that any order polytope of dimension $12$ or $13$ is Ehrhart positive. This solves an open problem proposed by Liu and Tsuchiya. Besides, we also verify that any $h^{*}$-polynomial of order polytope of dimension $d\leq 13$ is real-rooted. |
| title | Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive |
| topic | Combinatorics Primary 05A15, Secondary 06A07, 68R05 |
| url | https://arxiv.org/abs/2412.07164 |