Permutations of a Set and the Set of Permutations of [n]

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $A$ be a set, let $n\in\mathbb{N}_{0}$, and let $[n]$ be as in Intervals of Natural Numbers §segment.

¶ A permutation of $A$ is a bijection $\sigma:A\to A$.

¶ $S_{n}=\{\sigma\in[n]^{[n]}:\sigma\text{ is bijective}\}$ is the set of permutations of $[n]$.