An Upper Bound on Stirling Number of the Second Kind

We shall show an upper bound on the Stirling number of the second kind, a byproduct of a homework exercise of Probabilistic Combinatorics offered by Prof. Tom Bohman.

Definition. A Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S(n,k).

Proposition. For all n, k, we have S(n,k) \leq \frac{k^n}{k!}\left(1-(1-1/m)^k\right)^m.

Proof. Consider a random bipartite graph with partite sets U:=[n], V:=[k]. For each vertex u\in U, it (independently) connects to exactly one of the vertices in V uniformly at random. Suppose X is the set of non-isolated vertices in V. It is easy to see that \operatorname{Pr}\left(X=V\right) = \frac{\text{number of surjections from }U\text{ to }V}{k^n} = \frac{k!S(n,k)}{k^n}.

On the other hand, we claim that for any \emptyset \neq A \subset [k] and i \in [k]\setminus A, \operatorname{Pr}\left(i\in X \mid A\subset X\right) \leq \operatorname{Pr}\left(i\in X\right). Note that the claim is equivalent to \operatorname{Pr}\left(A\subset X \mid i\notin X\right) \geq \operatorname{Pr}\left(A\subset X\right). Consider the same random bipartite graph with V replaced by V':=[k]\setminus \{i\} and let X' be the set of non-isolated vertices in V'. The claim is justified since \operatorname{Pr}\left(A\subset X\mid i\notin X\right) = \operatorname{Pr}\left(A\subset X'\right) \geq \operatorname{Pr}\left(A\subset X\right).

Set A:=[i-1] in above for i = 2, \ldots, k. Using the multiplication rule with telescoping the conditional probability, we obtain \begin{aligned}\operatorname{Pr}\left(X=V\right) =& \operatorname{Pr}\left(1\in X\right)\operatorname{Pr}\left(2\in X \mid [1]\subset X\right) \\ & \ldots \operatorname{Pr}\left(k\in X\mid [k-1]\subset X\right)\\ \leq & \operatorname{Pr}\left(1\in X\right)\operatorname{Pr}\left(2\in X\right)\ldots\operatorname{Pr}\left(k\in X\right) \\ = & \left(1-(1-1/m)^k\right)^m.\end{aligned} [qed]