On the probability of coincidence of cycle lengths for independent random permutations with given number of cycles

From the set of all permutations of the degree $n$ with a given number $ N \le n $ of cycles two permutations are choosed randomly, uniformly and independently. The cycles of each permutation are numbered in some of $N!$ possible ways. We study the coincidence probability of the cycle lengths of permutations for a given numbering. This probability up to a suitably selected renumbering of cycles of the first permutation equals to the probability of similarity of these permutations. The asymptotic estimates of the coincidence probability of the cycle lengths are obtained for five types of relations between $N,n\to\infty$.