Permutations with equal orders. (27th September 2021) - Acan, Huseyin; Burnette, Charles; Eberhard, Sean; Schmutz, Eric; Thomas, James
Abstract: Let P(ordπ = ordπ ') be the probability that two independent, uniformly random permutations of [ n ] have the same order. Answering a question of Thibault Godin, we prove that P(ordπ = ordπ ') = n^ - 2 + o(1) and that P(ordπ = ordπ ') 1 2n^ - 2lg*n for infinitely many n . (Here lg * n is the height of the tallest tower of twos that is less than or equal to n .)