TITLE: A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$
QUESTION [14 upvotes]: $\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is: 
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general formula (or a nice combinatorial description) for the number of conjugacy classes produced in this case?
Some thoughts on the problem: for a fixed $n$ one can compute this by hand as follows:
Note that if we consider the action of $\S_n \times \S_n$ on $\S_n \times \S_n$ by conjugation, then we have classes of types $(\lambda, \mu)$ for each configuration $\lambda$ and $\mu$ of the Young diagram. Here we restrict ourselves to the diagonal action, so this would split the $(\lambda,\mu)$ pair further. To compute this splitting for a fixed pair $(x,y)$ where $x$ is of type $\lambda$, y is of type $\mu$, let $C(x)$ denotes the centralizer of $x$. For any $g \in \S_n$, write $g = g_1g_x$ for some $g_x \in C(x)$. Then
$(x,y) \sim (g_1xg_1^{-1}, g_1(g_xyg_x^{-1})g_1^{-1})$
Then the number of equivalence classes we have for this pair is the number of orbits of the class $\mu$ acted upon by $C(x)$ via conjugation.
But is there a general formula?
Thanks,
Ngoc

REPLY [3 votes]: I discovered this old question in connection with someone else's similar (and current) question: The Simultaneous Conjugacy Problem in the symmetric group $S_N$
If anyone still cares, here is a slightly different answer for your question using the character table instead of centralizer sizes.  The characters might be more accessible in certain cases (and of interest in their own right).
$$N=\frac{1}{n!} \sum_{\alpha \in G} \left( \sum_{\text{irreps}\ \chi} \chi(\alpha)^2 \right)^2$$
The inner sum is just the centralizer term in Marty Isaacs answer, but expressed in a different way.  It might be of some use in this form.  For example, if you want to enrich the group action to allow coordinate swap or inversion, similar class functions can be used.
The OP and others may be interested in the above linked question, which discusses the related decision problem and the generalization to more coordinates.