Random partitions of sets with a known number of blocks

We consider the class of all partitions of a set of $n$ elements into $N$ blocks. Provided that the uniform distribution is given on this class and $n,N\to\infty$, we describe the asymptotic behaviour of the mathematical expectation and variance and prove Poisson and local normal limit theorems for the random variable equal to the number of blocks of a given size in a partition chosen at random. We find asymptotic expansions of the number of partitions of a set of $n$ elements into $N$ blocks with exactly $k=k(n,N)$ blocks of a given size.