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?
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$\begingroup$ There are many results, so what do you need? $\endgroup$– Fedor PetrovCommented Nov 18 at 7:03
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1$\begingroup$ Essentially a repeat of mathoverflow.net/questions/62088. $\endgroup$– Richard StanleyCommented Nov 18 at 11:42
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$\begingroup$ I don't know if there are some estimates or specific properties on these numbers just combinatorically. Or it is just too difficult to calculate directly? $\endgroup$– user545662Commented Nov 19 at 5:08
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$\begingroup$ There are a vast number of papers on special cases, one example being Goupil's paper at sciencedirect.com/science/article/pii/0012365X9090054L. I wouldn't be surprised if the general problem is #P-complete. Maybe this is already known. See also inria.hal.science/inria-00098749/document. $\endgroup$– Richard StanleyCommented Nov 20 at 1:07
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