Definition:Permutation

Mapping
A bijection $f: S \to S$ from a set $S$ to itself is called a permutation on (or of) $S$.

Ordered Selection
Let $S$ be a set of $n$ elements.

Let $r \in \N: r \le n$.

An $r$-permutation of $S$ is an ordered selection of $r$ elements of $S$.

This can denoted $P_{nr}$, ${}^r P_n$, ${}_r P_n$ or even (extra confusingly) ${}_n P_r$ (there is little consistency in the literature).

From this definition, it can be seen that a bijection $f: S \to S$ (as defined above) is an $n$-permutation.

From Number of Permutations it can be seen that:
 * $P_{nr} = \dfrac {n!} {\left({n-r}\right)!}$
 * $P_{nn} = n!$

Using the falling factorial symbol, this can also be expressed:
 * $P_{nr} = n^{\underline r}$

Also see

 * Permutation on n Letters

Linguistic Note
As Don Knuth points out, Vaughan Pratt has made the suggestion that, because permutations are so important in the field of computer science, they be called perms:
 * "As soon as Pratt's convention is established, textbooks of computer science will become somewhat shorter (and perhaps less expensive)."