Coalescence Probabilities of Cycle Products

TL;DR

This paper generalizes Stanley's formula to compute the probability that a set of elements are in the same cycle of a product of two random n-cycles, providing a combinatorial proof and explicit formula.

Contribution

It offers a new combinatorial proof and explicit formula for coalescence probabilities in products of two random n-cycles, extending Stanley's original result.

Findings

Derived an explicit formula for cycle coalescence probabilities

Provided a combinatorial proof of the generalized formula

Extended Stanley's formula to broader cases

Abstract

Generalizing a formula of Stanley, we prove combinatorially that the probability that $1, 2, \dots, k$ are contained in the same cycle of a product of two random $n$-cycles is \[\frac{1}{k} + \frac{4 (-1)^n}{ \binom{2k}{k}} \sum_{\substack{1 \leq i \leq k-1 \\ i \not\equiv n \bmod 2}} \binom{2k-1}{k+i} \left(\frac{1}{n+i+1} - \frac{1}{n-i}\right).\]