On the rate of convergence of the distribution of the number of cycles of given length in a random permutation with known number of cycles to the limit distributions

Let on the set $S_{n,N}$ of all different permutations of degree $n$ with $N$ cycles the uniform distribution be given. We obtain estimates of the rate of convergence of the distribution of the number of cycles of given length in a random permutation of $S_{n,N}$ to the limit distributions as $n,N\to\infty$ in such a way that either $n/N\to 1$ or $n/N\to\infty$.