The Rényi entropy of the order of a random permutation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917496154488832 |
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| author | Beker, Adrian |
| author_facet | Beker, Adrian |
| contents | We study the distribution of the order of a random permutation of $[n]$ through the lens of Rényi entropy. In particular, we obtain an asymptotic for the Rényi $q$-entropy of the order in the full range $1 \leq q \leq \infty$. For $q > 1$, our results are quantitatively optimal and reveal a tight connection between the asymptotic behaviour of the Rényi $q$-entropy and arithmetic properties of $n$. Of particular interest are the cases $q = \infty$ and $q = 2$, which correspond to the maximum probability of achieving a particular order and the probability that two independent random permutations have equal orders, respectively. In the former case, we show that the probability in question is asymptotic to $1/n$ and additionally characterise the maximiser for sufficiently large $n$. In the latter case, we determine a minimal and maximal order for the probability as a function of $n$, of respective forms $c/n^2$ and $\log^*n/n^2$. Our results provide an essentially complete answer to a set of questions raised by Acan, Burnette, Eberhard, Schmutz and Thomas, some of which go back to work of Erdős and Turán from the 1960s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Rényi entropy of the order of a random permutation Beker, Adrian Combinatorics Group Theory Number Theory We study the distribution of the order of a random permutation of $[n]$ through the lens of Rényi entropy. In particular, we obtain an asymptotic for the Rényi $q$-entropy of the order in the full range $1 \leq q \leq \infty$. For $q > 1$, our results are quantitatively optimal and reveal a tight connection between the asymptotic behaviour of the Rényi $q$-entropy and arithmetic properties of $n$. Of particular interest are the cases $q = \infty$ and $q = 2$, which correspond to the maximum probability of achieving a particular order and the probability that two independent random permutations have equal orders, respectively. In the former case, we show that the probability in question is asymptotic to $1/n$ and additionally characterise the maximiser for sufficiently large $n$. In the latter case, we determine a minimal and maximal order for the probability as a function of $n$, of respective forms $c/n^2$ and $\log^*n/n^2$. Our results provide an essentially complete answer to a set of questions raised by Acan, Burnette, Eberhard, Schmutz and Thomas, some of which go back to work of Erdős and Turán from the 1960s. |
| title | The Rényi entropy of the order of a random permutation |
| topic | Combinatorics Group Theory Number Theory |
| url | https://arxiv.org/abs/2605.15063 |