$c$-Birkhoff polytopes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913039396110336 |
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| author | Banaian, Esther Chepuri, Sunita Gunawan, Emily Pan, Jianping |
| author_facet | Banaian, Esther Chepuri, Sunita Gunawan, Emily Pan, Jianping |
| contents | In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a $c$-Birkhoff polytope for each Coxeter element $c$ of the symmetric group. We then show that the $c$-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the $c$-sorting word of the longest permutation. When $c=s_1s_2\dots s_{n}$, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the $c$-Birkhoff polytope is the number of the longest chains in the (type A) $c$-Cambrian lattice. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $c$-Birkhoff polytopes Banaian, Esther Chepuri, Sunita Gunawan, Emily Pan, Jianping Combinatorics 52B20, 05A05, 06A07 In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a $c$-Birkhoff polytope for each Coxeter element $c$ of the symmetric group. We then show that the $c$-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the $c$-sorting word of the longest permutation. When $c=s_1s_2\dots s_{n}$, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the $c$-Birkhoff polytope is the number of the longest chains in the (type A) $c$-Cambrian lattice. |
| title | $c$-Birkhoff polytopes |
| topic | Combinatorics 52B20, 05A05, 06A07 |
| url | https://arxiv.org/abs/2504.07505 |