Stack-sorting simplices: geometry and lattice-point enumeration
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Lee, Eon Mitchell, Carson Vindas-Meléndez, Andrés R. |
| author_facet | Lee, Eon Mitchell, Carson Vindas-Meléndez, Andrés R. |
| contents | We initiate the study of subpolytopes of the permutahedron that arise as the convex hulls of stack-sorting on permutations. We primarily focus on $Ln1$ permutations, i.e., permutations of length $n$ whose penultimate and last entries are $n$ and $1$, respectively. First, we present some enumerative results on $Ln1$ permutations. Then we show that the polytopes that arise from stack-sorting on $Ln1$ permutations are simplices and proceed to study their geometry and lattice-point enumeration. In addition, we pose questions and problems for further investigation. Particular focus is then taken on the $Ln1$ permutation $23\cdots n1$. We show that the convex hull of all its iterations through the stack-sorting algorithm shares the same lattice-point enumerator as that of the $(n-1)$-dimensional unit cube and lecture-hall simplex. Lastly, we detail some results on the real lattice-point enumerator for variations of the simplices arising from stack-sorting on the permutation $23\cdots n1$. This then allows us to show that those simplices are Gorenstein of index $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16457 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stack-sorting simplices: geometry and lattice-point enumeration Lee, Eon Mitchell, Carson Vindas-Meléndez, Andrés R. Combinatorics 52B05, 05A15, 52B20 We initiate the study of subpolytopes of the permutahedron that arise as the convex hulls of stack-sorting on permutations. We primarily focus on $Ln1$ permutations, i.e., permutations of length $n$ whose penultimate and last entries are $n$ and $1$, respectively. First, we present some enumerative results on $Ln1$ permutations. Then we show that the polytopes that arise from stack-sorting on $Ln1$ permutations are simplices and proceed to study their geometry and lattice-point enumeration. In addition, we pose questions and problems for further investigation. Particular focus is then taken on the $Ln1$ permutation $23\cdots n1$. We show that the convex hull of all its iterations through the stack-sorting algorithm shares the same lattice-point enumerator as that of the $(n-1)$-dimensional unit cube and lecture-hall simplex. Lastly, we detail some results on the real lattice-point enumerator for variations of the simplices arising from stack-sorting on the permutation $23\cdots n1$. This then allows us to show that those simplices are Gorenstein of index $2$. |
| title | Stack-sorting simplices: geometry and lattice-point enumeration |
| topic | Combinatorics 52B05, 05A15, 52B20 |
| url | https://arxiv.org/abs/2308.16457 |