On the distribution of the number of cycles of a given length in the class of permutations with known number of cycles

We consider the set of all permutations of degree $n$ with $N$ cycles. We assume that the uniform distribution is defined on this set and consider the random variable equal to the number of cycles of a given length in the random permutation from this set. We obtain the asymptotic values of the mathematical expectation and the variance of this random variable and prove the limit theorems on the convergence to the Poisson and the Gaussian distributions as $n,N\to\infty$. We give the asymptotic expansions for the number of permutations of degree $n$ with $N$ cycles among which there are exactly $k=k(n,N)$ of a given length.