Counting permutation cycle types in $S_n$

Question

Consider the sym group $S_4$,

We write the elements of $S_4$ in a given "form" such as \begin{align*} e = (1)(2)(3)(4) &\sim 1,1,1,1 \\ (12)(34) &\sim 2,2 \\ (123)(4) &\sim 3,1 \end{align*}

Note that the number indicate the lenght of a cycle.

Also, notice that $(123)(4)$, $(124)(3)$ and $(234)(1)$ (also their inverses) are in the form of $3,1$

So here are the list for $S_4$: \begin{align*} &1,1,1,1\\ &2,1,1 \\ &2,2 \\ &3,1 \\ &4 \end{align*}

There are $5$.

For $S_5$: \begin{align*} &1,1,1,1,1 \\ &2,1,1,1 \\ &2,2,1, \\ &3,1,1 \\ &3,2 \\ &4,1 \\ &5 \end{align*}

There are $7$.

For $S_6$, there are $11$.

Now, I'm trying to make a fomula or a way to determine how many "forms" in $S_n$, for all $n$. However, I can't figure it out..

Please help me with this problem I just made.

Thank you very much.

Answer 1

Note that cycle types in $S_n$ can be viewed as unordered integer partitions of $n$ and vice versa, hence the answer is $p\left(n\right)$ (see here). Unfortunately there's no nice formula for the output of the partition function, but its generating function is quite nice and it is easy to look up values for small $n$.