Limit Values of Character Sums in Frobenius Formula of Three Permutations

Abstract

We study the asymptotic behaviour of the character summation part of the Frobenius formula for three conjugacy classes of the symmetric groups. When two conjugacy classes contain fixed points of order $\sim H_i\sqrt{n}$ for $i=1,2$ and all other cycles are long, the summation converges uniformly to $2e^{-H_1{H}_2}$; the exponential factor coincides with the non-collision probability of the birthday paradox. When short cycles are rare in all three classes, character sum tends to $2$.