Given a set of $N$ labeled elements $\{1, 2, ..., n\}$, we know that there are $S(N, k)$ ways to partition the elements into $k$ non-empty subsets (where $S(N, k)$ is the Stirling number of the second kind), and $B_n$ total ways to partition the elements into any number of subsets (where $B_n$ is the $n$-th Bell number).
In Chern et al. (2014), the authors give a nice expression for the number of blocks of size $i$ in the set of all partitions:
$$M(X_i;n) = {{n}\choose{i}} B_{n-i}$$
Is there a similar expression for the number of blocks of size $i$ in the set of partitions with $k$ subsets $M(X_i;n, k)$?
Yes. The same idea can be applied: Choose $i$ of $n$ elements and partition the remaining $n-i$ elements into $k-1$ subsets, yielding
$$ \binom niS(n-i,k-1)\;. $$