Tootfinder

The most probable order of a random permutation
Adrian Beker
https://arxiv.org/abs/2510.11698 https://arxiv.org/pdf/2510.11698

The most probable order of a random permutation
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, Eberh…

Courtesy of @…, this week’s #ThursdayFiveList theme is #RainyDayMusic:
1. Corey Glover, “April Rain”—

#Wordle 1,584 5/6*
⬜⬜⬜⬜🟨 <1% of 213,807 (150)
🟨🟨🟨⬜⬜ 0 of 29 (3)
⬜🟨🟨🟨⬜ 0 of 0 (1)
⬜🟩🟨🟩🟩 0 of 0 (1)
🟩🟩🟩🟩🟩
WordleBot
Skill 63/99
Luck 59/99
Killer wordle for me today. 1st user error on guess 3, then I could not think of ANY word for guess 4, was trying things at random, and it took what I would have thought was a proper name, and that got me the answer in guess 5!
Looked up guess 4 found other meanings, still not a wordle word.