Let us consider the set of all $n$ bit strings $\{0, 1\}^{n}$. Consider a partition of this set $\{S_{1}, S_{2}, \cdots, S_{k}\}$ for a fixed $k$ (such that $S_{1} \cup S_{2} \cup \cdots S_{k} = \{0, 1\}^{n}$.)
How many ways can we choose $\{S_{1}, S_{2}, \cdots, S_{k}\}$ (all non-empty partitions) such that there is at least one $i \in [k]$ for which the number of even numbers in $S_{i}$ is exactly $\frac{2}{3}|S_{i}|$?
I know that the number of ways I can choose $k$ non-empty partitions is related to Sterling number's of the second kind. But I cannot incorporate the second constraint (that of one partition containing a certain number of even numbers) into my analysis.