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:
 * $$P_{nr} = \frac {n!} {\left({n-r}\right)!}$$;
 * $$P_{nn} = n!$$.

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. To quote: "As soon as Pratt's convention is established, textbooks of computer science will become somewhat shorter (and perhaps less expensive)."