Is there a formula for the number of equipartitions of $[n]$ into $k$ parts of size $s = n/k$?

Question

Let $k$ divide $n$ and $Q(n,k)$ be the number of partitions of $n$ into $k$ parts, each of which has size $s = n/k$.

Is there a formula for $Q(n,k)$?

What is the asymptotic behavior of $Q(n,k(n))$ as $n, k(n) \to \infty$?

Answer 1

The Multinomial Coefficient includes $\binom{n}{k_1,\cdots,k_r}=\frac{n!}{k_1!k_2!\cdots k_n!}$. This gives the number of ways to divide $n$ into $r$ sets consisting of $k_1$, $k_2$, $\cdots$, $k_r$ elements.

In your case, you are looking at $r=k$ and $k_i=s$ because all of the partitions are the same size. Therefore, there are $$ \binom{n}{s,\cdots,s}=\frac{n!}{(s!)^k} $$ ways to partition $n$ into $k$ labeled sets of size $s$. Since all the sets are the same size, they are interchangeable, and we must divide by $k!$ for all the rearrangements.

Therefore, $$ Q(n,k)=\frac{1}{k!}\binom{n}{s,\cdots,s} $$