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Claim. Problem 1161 asks for which $k$ the number $f_k(n)$ of permutations in $S_n$ of order exactly $k$ is largest. Writing $p_n(m)=f_m(n)/n!$ for the probability that a uniform random permutation of $n$ letters has order $m$, and $M(n)$ for the largest value of $p_n(m)$ over $m$, Beker proves two theorems. Theorem 1.1: $M(n)\sim 1/n$, which is the probability of an $n$-cycle, and for all sufficiently large $n$ every $m$ with $p_n(m)\geq 1/n$ has the form $m=n-j$ with $j$ in

$$K_n=\{\,j\in\{0,1,\ldots,n-1\} : \operatorname{lcm}(1,2,\ldots,j)\mid n-j\,\}.$$

Theorem 1.2: for all sufficiently large $n$, $p_n(m)=M(n)$ holds if and only if $m=n-\max K_n$. In the problem's terms, for every large $n$ the count $f_k(n)$ is maximal at exactly one order, the least $k\geq1$ divisible by every integer from $1$ to $n-k$, and the maximum is $(1+o(1))\,(n-1)!$. The paper answers the question of Erdős and Turán (1968) that the problem records, as restated by Acan, Burnette, Eberhard, Schmutz and Thomas.

Qualification. The identification of the maximizing order holds for all sufficiently large $n$. Remark 1.3 says that the bound one could extract from the argument is most probably not small enough to check the remaining cases by a naive method, and that the hypothesis cannot be dropped, since small $n$ have other maximizers. The exact question for every $n$ is therefore settled only from an unspecified point on; the site's label counts this as a solution, and the acceptance recorded below is the curator's.

Proof shape. Theorem 1.1 studies the joint distribution of the order and the number of cycles and applies a different local limit law in each of the large, intermediate and small cycle-count ranges, avoiding lower tail bounds for the number of cycles, with the collision-entropy method of Acan and coauthors refined for the purpose. Theorem 1.2 adds a local limit law (Proposition 4.1) showing that the probability of order $n-j$, for $j\in K_n$, is $1/(n-j)$ up to a small error, which decides the comparison among the candidates. The source card is beker_2025_most_probable_order_random_permutation; nothing on this page is independently reviewed by this project.

Source. Adrian Beker, The most probable order of a random permutation, arXiv:2510.11698, version 1 submitted 2025-10-13, 8 pages. The arXiv record gives no journal reference.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, marks Problem 1161 solved and credits Beker's paper (problem page last edited 1 February 2026), after a discussion-thread comment of 2026-01-24 pointed the site at the preprint. No refereed publication is recorded, so the evidence is the curator's acceptance alone.