As an example, suppose we have the set $\{1,2,3,4,5\}$ and say I want a partition with one singleton and two two-element sets. I could then have $1|23|45$, $2|13|45$, etc. Also, $1|23|54|$ or $1|45|23|$ are not counted as separate from $1|23|45|$. I would like to count the number of these.
I know that in general, if we are looking at a partition $P$ of $n$, where $P=(a_1,b_1),(a_2,b_2),...,(a_k,b_k)$ (is this the right notation?) where $a_1b_1+a_2b_2+\cdots+a_kb_k=n$ (so $b_i$ is the number of times $a_i$ shows up in the partition), then the number I am looking for is $$\frac{1}{b_1!b_2!\cdots b_k!}\binom{n}{a_1}\binom{n-a_1}{a_1}\cdots\binom{n-a_1(b_1-1)}{a_1}\binom{n-a_1b_1}{a_2}\cdots \binom{n-a_1b_1-a_2b_2-\cdots-a_k(b_k-1)}{a_k}$$
Is there anything more known about such numbers, or an easier way to write this expression? Any reference would be super helpful. Thanks!