In how many ways can we partition $k$ (distinct) items into $n$ blocks so that we have $k_i$ blocks of size $i$ for each $i$?

Question

This question originates from Problem 139 at https://bogart.openmathbooks.org/ctgd/s3-2-partitions.html, which I located a solution file at

https://github.com/OpenDiscreteMath/ibl-combinatorics/blob/update-bogart/ComboNoteswSolutions11-6-04.pdf

The solution, however, appears to contain some serious typos:

In particular, as I understand it, all "$n$" in the Solution section should be replaced by "$k$", resulting in what I think is the correct solution:

\begin{align*} \frac{k!}{\prod_{i=1}^k(i!)^{k_i}k_i!} \end{align*}

Is my understanding correct?

Answer 1

Your formula is correct; a proof is given e.g. by Andrews in The Theory of Partitions, Theorem 13.2: