My question is that I was working with some counting problems, and finally the answer should be $$ \nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\mu_1},\sigma_2\in C_{\mu_2},\sigma_3\in C_{\mu_3}\}. $$ $C_{\mu_i}$ are conjugacy classes in $S_n$. $\sigma_i \in S_n$ are the permutations. With little work we can show $$ \nu_{\mu_1,\mu_2,\mu_3}=\frac{(n!)^2}{c(\mu_1)c(\mu_2)c(\mu_3)}\cdot \sum_{\chi\in \mathrm{Irr}(S_n)} \frac{\chi(\mu_1)\chi(\mu_2)\chi(\mu_3)}{\chi(1)}. $$ It's called class multiplication coefficients. Are there any results on these numbers?