Given an element $\sigma \in S_n$ , I wish to calculate number of elements $\gamma \in S_n$ such that $\sigma \circ \gamma \circ \sigma^{-1} = \gamma $. I suspect the number to be $$\frac{n!}{1^{n_1}n_1!2^{n_2}n_2!3^{n_3}n_3!\ldots n^{n_n}n_n!}$$ where $(n_1,n_2,n_3,\ldots,n_n)$ is the cycle type of $\sigma$. However I am not able to prove this. Any help would be appreciated. Thanks for reading.
Edit: As suggested in the comment orbit-stabiliser theorem is useful. Also I notice that conjugating does not change cycle type. Thus size of orbit is $$\frac{n!}{1^{n_1}n_1!2^{n_2}n_2!3^{n_3}n_3!\ldots n^{n_n}n_n!}$$ so by orbit stabiliser theorem the cardinality of set of fixed points is $${1^{n_1}n_1!2^{n_2}n_2!3^{n_3}n_3!\ldots n^{n_n}n_n!}$$