Limit Values of Character Sums in Frobenius Formula of Three Permutations

We study the asymptotic behavior of the character sums appearing in the Frobenius formula for three conjugacy classes of symmetric groups. We show that if all three conjugacy classes contain no cycles of lengths $1$, $2$, or $3$, then the character sum converges to $2$. On the other hand, if two of the conjugacy classes contain $H\sqrt{n}$ fixed points while all other cycle lengths in all three conjugacy classes are large, then the character sum converges to $2e^{-H^2}$. As consequences of these results, we obtain several corollaries and propose conjectures related to the Hurwitz problem with three branching points on the sphere.