  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 $\Gamma_2(A_n)$.
