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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2406.19022 |
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| _version_ | 1866910504170029056 |
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| author | Meshulam, Roy Moyal, Omer |
| author_facet | Meshulam, Roy Moyal, Omer |
| contents | Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $π\in \mathbb{S}_n$ let $X_π$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all $\{i_0,\ldots, i_m\} \subset [n]$ such that $i_0<\cdots<i_m$ and $π(i_0)<\cdots < π(i_m)$. For $r \geq 0$ let $p_r(n)$ denote the probability that $X_π$ is not topologically $r$-connected for $π\in \mathbb{S}_n$. It is shown that for fixed $r \geq 0$ there exist constants $0<C_r, C_r' < \infty$ such that \[ C_r \frac{(\log n)^r}{n} \leq p_r(n) \leq C_r' \frac{(\log n)^{2r}}{n}. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological connectivity of random permutation complexes Meshulam, Roy Moyal, Omer Combinatorics 05E45, 60C05 Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $π\in \mathbb{S}_n$ let $X_π$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all $\{i_0,\ldots, i_m\} \subset [n]$ such that $i_0<\cdots<i_m$ and $π(i_0)<\cdots < π(i_m)$. For $r \geq 0$ let $p_r(n)$ denote the probability that $X_π$ is not topologically $r$-connected for $π\in \mathbb{S}_n$. It is shown that for fixed $r \geq 0$ there exist constants $0<C_r, C_r' < \infty$ such that \[ C_r \frac{(\log n)^r}{n} \leq p_r(n) \leq C_r' \frac{(\log n)^{2r}}{n}. \] |
| title | Topological connectivity of random permutation complexes |
| topic | Combinatorics 05E45, 60C05 |
| url | https://arxiv.org/abs/2406.19022 |