I'm working on estimating the error probability of some Maximum Likelihood Estimator, where the problem boils down to obtaining upper and lower bounds of a combinatorial number described below.
Suppose I have $n$ distinct balls, and I want to assign them into $k$ disjoint groups. Each ball has to be assigned to exactly one group, and each group must contain at least one ball. Denote the number of all possible such assignments by $S(n, k)$. Is there any existing results about the asymptotic behavior of $S(n, k)$ as $n$ tends to infinity while $k$ remains fixed?