Coalescence Probabilities of Cycle Products
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908284645015552 |
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| author | Mui, Holden |
| author_facet | Mui, Holden |
| contents | Generalizing a formula of Stanley, we prove combinatorially that the probability that $1, 2, \dots, k$ are contained in the same cycle of a product of two random $n$-cycles is \[\frac{1}{k} + \frac{4 (-1)^n}{ \binom{2k}{k}} \sum_{\substack{1 \leq i \leq k-1 \\ i \not\equiv n \bmod 2}} \binom{2k-1}{k+i} \left(\frac{1}{n+i+1} - \frac{1}{n-i}\right).\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coalescence Probabilities of Cycle Products Mui, Holden Combinatorics Generalizing a formula of Stanley, we prove combinatorially that the probability that $1, 2, \dots, k$ are contained in the same cycle of a product of two random $n$-cycles is \[\frac{1}{k} + \frac{4 (-1)^n}{ \binom{2k}{k}} \sum_{\substack{1 \leq i \leq k-1 \\ i \not\equiv n \bmod 2}} \binom{2k-1}{k+i} \left(\frac{1}{n+i+1} - \frac{1}{n-i}\right).\] |
| title | Coalescence Probabilities of Cycle Products |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.01415 |