Consider a product of cycles of the form $(\underline{6})(\underline{2})(\underline{2})$ in $S_9$ where $(\underline{i})$ is a cycle with length $i$. I want to determine the number of such cycles.
However, in order for $(\underline{6})(\underline{2})(\underline{2})$ to even make sense, it must be the case that $(\underline{2})(\underline{2})$ is a product of transpositions. So if we fix $a = 1,\dots,9$, it follows that such a product can be written as $$(ax)(ay) = (ayx)$$ where $x,y = 1,2,\dotsc,9$ and $x\neq y$. Therefore, the product $(\underline{6})(\underline{2})(\underline{2}) = (\underline{6})(\underline{3})$, so it follows that there are $$\frac{9!}{18} = 20,160$$ such elements of that form.
Is the above calculation correct? I feel as if I overcounted.