Problem 1161

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claims/: The 1 claim page of Problem 1161, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f_k(n)$ count the number of elements of $S_n$ of order $k$. For which values of $k$ will $f_k(n)$ be maximal?

Formulation. The question is read for large $n$, as the paper the site credits and the site's label read it. Beker takes up the question of Erdős and Turán (1968, p. 414), restated by Acan, Burnette, Eberhard, Schmutz and Thomas, as one about the most probable order of a random permutation, and calls his answer, which holds for all sufficiently large $n$, essentially complete. Read for every $n$, the question is settled only beyond a threshold the argument does not make explicit, and some small $n$ have other maximizers (Remark 1.3). The page's standing targets the large-$n$ reading.

Status. Solved.

Source. erdosproblems.com/1161, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1161, https://www.erdosproblems.com/1161.

References.

• [Be25d] A. Beker, The most probable order of a random permutation. arXiv:2510.11698 (2025).

Formalization. None recorded.

Current assessment

The standing judges the site's formulation of 2026-09-04, read for large $n$ as the Formulation states. Beker [Be25d] answers it for all sufficiently large $n$: the largest of the counts $f_k(n)$ is $(1+o(1))\,(n-1)!$, and it is attained at exactly one order, the least $k\geq1$ divisible by every integer from $1$ to $n-k$; the paper's own Remark 1.3 says that small $n$ behave differently and that the bound its argument could give is most probably not small enough to check the remaining cases by a naive method. The accepted claim page Beker 2025 carries the acceptance evidence, which is the site's curator crediting the preprint as the solution; no refereed publication is recorded. Search scope: the site's problem page, discussion thread and proof-claims page, and the arXiv record of the preprint, accessed 2026-10-07; they record no other claim. Nothing on this page is independently reviewed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• beker_2025_most_probable_order_random_permutation
• beker_2025_most_probable_order_random_permutation / theorem_1_1
• beker_2025_most_probable_order_random_permutation / theorem_1_2