I want to know if there is some systematic way (using some combinatorial argument) to find the number of elements of conjugacy classes of $S_n$ for some given $n$.
For example, let's consider $S_5$. If the representative for the conjugacy class is an $m$-cycle then Dummit and Foote gives a formula on how to compute the number of elements in the conjugacy class. This is not a problem. But what about when the representative is not an $m$-cycle. As an example we can consider the conjugacy class that gives rise by the partition $2+3$ of $5$. A representative for the conjugacy class would be $(1 2)(3 4 5)$. How can I find the number of such elements?.
Question?:
Does $ {5\choose 2}\cdot { 3 \choose 3}\cdot 2$ give me what I want?
Reasoning: For the first parenthesis I need to choose $2$ elements out of $5$ and for the second set of parenthesis I need to choose $3$ out of the remaining $3$ (noting that they can't be repeats). Finally we can permute these two parenthesis in two ways, thus giving me the above number.
Is this reasoning correct?. If not how does one find the number of elements of such conjugacy classes.
As always, any help is greatly appreciated.