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