Separation probabilities for products of permutations
  We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of the random permutation of $\{1,2,...,n\}$ obtained as product of two uniformly random $n$-cycles. 
Olivier Bernardi (MIT), Alejandro H. Morales (MIT), Richard P. Stanley   (MIT), Rosena R. X. Du
math.CO