The most probable order of a random permutation
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908589781680128 |
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| author | Beker, Adrian |
| author_facet | Beker, Adrian |
| contents | Given positive integers $n$ and $m$, let $p_n(m)$ be the probability that a uniform random permutation of $[n]$ has order exactly $m$. We show that, as $n \to \infty$, the maximum of $p_n(m)$ over all $m$ is asymptotic to $1/n$, the probability of an $n$-cycle. Furthermore, for sufficiently large $n$, we show that the maximum is attained precisely if $m$ is the least positive integer divisible by all positive integers less than or equal to $n-m$. This answers a question of Acan, Burnette, Eberhard, Schmutz and Thomas, originally attributed to work of Erdős and Turán from 1968. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The most probable order of a random permutation Beker, Adrian Combinatorics Group Theory Given positive integers $n$ and $m$, let $p_n(m)$ be the probability that a uniform random permutation of $[n]$ has order exactly $m$. We show that, as $n \to \infty$, the maximum of $p_n(m)$ over all $m$ is asymptotic to $1/n$, the probability of an $n$-cycle. Furthermore, for sufficiently large $n$, we show that the maximum is attained precisely if $m$ is the least positive integer divisible by all positive integers less than or equal to $n-m$. This answers a question of Acan, Burnette, Eberhard, Schmutz and Thomas, originally attributed to work of Erdős and Turán from 1968. |
| title | The most probable order of a random permutation |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2510.11698 |