The variance of the Stirling cycle numbers

TL;DR

This paper investigates the probability that two random permutations of n letters share the same number of cycles, revealing it asymptotically approaches 1 divided by twice the square root of pi times the logarithm of n.

Contribution

It provides a new asymptotic estimate for the probability related to cycle counts in permutations, advancing understanding of permutation cycle structure.

Findings

Probability asymptotically equals 1/(2√(π log n))

Shows the distribution of cycle counts in permutations

Enhances understanding of permutation cycle statistics

Abstract

We show that the probability that two permutations of $n$ letters have the same number of cycles is \[\sim \frac{1}{2\sqrt{\pi\log{n}}}.\]