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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.13877 |
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| _version_ | 1866910114168963072 |
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| author | Kong, Bochao Zeng, Ji |
| author_facet | Kong, Bochao Zeng, Ji |
| contents | The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This result has been established by Pawlowski in a more general form. Our proof is shorter, rather different, and gives an algebraic criterion for a non-standard permutohedron generated by $n$ distinct real numbers to require at least $n$ non-trivial hyperplanes to cover its vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | To cover a permutohedron Kong, Bochao Zeng, Ji Combinatorics The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This result has been established by Pawlowski in a more general form. Our proof is shorter, rather different, and gives an algebraic criterion for a non-standard permutohedron generated by $n$ distinct real numbers to require at least $n$ non-trivial hyperplanes to cover its vertices. |
| title | To cover a permutohedron |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.13877 |