A Vershik-Kerov theorem for wreath products
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915390143070208 |
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| author | Chatterjee, Sourav Diaconis, Persi |
| author_facet | Chatterjee, Sourav Diaconis, Persi |
| contents | Let $G_{n,k}$ be the group of permutations of $\{1,2,\ldots, kn\}$ that permutes the first $k$ symbols arbitrarily, then the next $k$ symbols and so on through the last $k$ symbols. Finally the $n$ blocks of size $k$ are permuted in an arbitrary way. For $σ$ chosen uniformly in $G_{n,k}$, let $L_{n,k}$ be the length of the longest increasing subsequence in $σ$. For $k,n$ growing, we determine that the limiting mean of $L_{n,k}$ is asymptotic to $4\sqrt{nk}$. This is different from parallel variations of the Vershik-Kerov theorem for colored permutations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Vershik-Kerov theorem for wreath products Chatterjee, Sourav Diaconis, Persi Probability Combinatorics 05A05, 60C05 Let $G_{n,k}$ be the group of permutations of $\{1,2,\ldots, kn\}$ that permutes the first $k$ symbols arbitrarily, then the next $k$ symbols and so on through the last $k$ symbols. Finally the $n$ blocks of size $k$ are permuted in an arbitrary way. For $σ$ chosen uniformly in $G_{n,k}$, let $L_{n,k}$ be the length of the longest increasing subsequence in $σ$. For $k,n$ growing, we determine that the limiting mean of $L_{n,k}$ is asymptotic to $4\sqrt{nk}$. This is different from parallel variations of the Vershik-Kerov theorem for colored permutations. |
| title | A Vershik-Kerov theorem for wreath products |
| topic | Probability Combinatorics 05A05, 60C05 |
| url | https://arxiv.org/abs/2408.04364 |