Coalescence Probabilities of Cycle Products

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 $$\frac{1}{k}+\frac{4(-1{)}^n}{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}\sum _{\begin{array}{c}1\le i\le k-1\\ i\equiv ̸n2\end{array}}\left(\genfrac{}{}{0px}{}{2k-1}{k+i}\right)(\frac{1}{n+i+1}-\frac{1}{n-i})\mathrm{.}$$