On random partitions

Let S be a finite set with n labeled elements. One of the partitions of S is selected at random each of them has the same probability. Harper determined the expected number of subsets in the random partition. Haigh studied the probability that the random partition has (at least one) subset of a given size. The present paper considers the expected number of subsets in a given size. Average and maximum are also determined. Results are generalized for the case if the number of subsets in the partition is also fixed.