Cycle lengths in a permutation are typically Poisson (Q870028)

Summary: The set of cycle lengths of almost all permutations in \(S_n\) are ``Poisson distributed'': we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain ``normal order'' (in the spirit of the Erdős-Turán theorem). Our results were inspired by analogous questions about the size of the prime divisors of ``typical'' integers.