Application of Character Estimates to the Number of $T_2$-Systems of the Alternating Group

We use character theory and character estimates to show that the number of $T_2$-systems of $A_n$ is at least \begin{equation*} \frac{1}{8n\sqrt{3}}\exp\left(\frac{2\pi}{\sqrt{6}}n^{1/2}\right)(1+o(1)). \end{equation*} Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph $\Gamma_2(A_n)$.