Partial permutohedra
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arXiv
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| Format: | Preprint |
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2022
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| author | Behrend, Roger E. Castillo, Federico Chavez, Anastasia Diaz-Lopez, Alexander Escobar, Laura Harris, Pamela E. Insko, Erik |
| author_facet | Behrend, Roger E. Castillo, Federico Chavez, Anastasia Diaz-Lopez, Alexander Escobar, Laura Harris, Pamela E. Insko, Erik |
| contents | Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is the convex hull of all vectors in $\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of $\mathcal{P}(m,n)$, and our methods and results include the following. For any $m$ and $n$, we obtain a bijection between the nonempty faces of $\mathcal{P}(m,n)$ and certain chains of subsets of $\{1,\dots,m\}$, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the $h$-polynomial of $\mathcal{P}(m,n)$. For any $m$ and $n$ with $n\ge m-1$, we use a pyramidal subdivision of $\mathcal{P}(m,n)$ to establish a recursive formula for the normalized volume of $\mathcal{P}(m,n)$, from which we then obtain closed expressions for this volume. We also use a sculpting process (in which $\mathcal{P}(m,n)$ is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of $\mathcal{P}(m,n)$ with arbitrary $m$ and fixed $n\le 3$, the normalized volume of $\mathcal{P}(m,4)$ with arbitrary $m$, and the Ehrhart polynomial of $\mathcal{P}(m,n)$ with fixed $m\le4$ and arbitrary $n\ge m-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_14253 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Partial permutohedra Behrend, Roger E. Castillo, Federico Chavez, Anastasia Diaz-Lopez, Alexander Escobar, Laura Harris, Pamela E. Insko, Erik Combinatorics 52B05, 52B11, 52B12, 52B20, 52A38 Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is the convex hull of all vectors in $\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of $\mathcal{P}(m,n)$, and our methods and results include the following. For any $m$ and $n$, we obtain a bijection between the nonempty faces of $\mathcal{P}(m,n)$ and certain chains of subsets of $\{1,\dots,m\}$, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the $h$-polynomial of $\mathcal{P}(m,n)$. For any $m$ and $n$ with $n\ge m-1$, we use a pyramidal subdivision of $\mathcal{P}(m,n)$ to establish a recursive formula for the normalized volume of $\mathcal{P}(m,n)$, from which we then obtain closed expressions for this volume. We also use a sculpting process (in which $\mathcal{P}(m,n)$ is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of $\mathcal{P}(m,n)$ with arbitrary $m$ and fixed $n\le 3$, the normalized volume of $\mathcal{P}(m,4)$ with arbitrary $m$, and the Ehrhart polynomial of $\mathcal{P}(m,n)$ with fixed $m\le4$ and arbitrary $n\ge m-1$. |
| title | Partial permutohedra |
| topic | Combinatorics 52B05, 52B11, 52B12, 52B20, 52A38 |
| url | https://arxiv.org/abs/2207.14253 |