The number of cycles in $S_{n}$ is $(n-1)!$. Thus its EGF is given by $\sum_{n=1}^{\infty} {(n-1)!} \frac{x^{n}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n}}{n!} = -\ln(1-x) $ The exponential formula yields that $\exp\left( \ln\left( \frac{1}{1-x} \right) \right) = 1+x^{2} + x^{3}+\cdots = \sum_{n=1}^{\infty} n! \frac{x^{n}}{n!}$where the right-hand is the EGF of $S_{n}$.