Definition:Permutation on n Letters

Definition
Let $\N^*_k$ be defined as the subset of natural numbers $\N^*_k = \left[{1 \,.\,.\, k}\right] = \left\{{1, 2, 3, \ldots, k}\right\}$.

A permutation on $n$ letters is a permutation $\pi: \N^*_n \to \N^*_n$.

The usual symbols for denoting a general permutation are $\pi$ (not to be confused with the famous circumference over diameter), $\rho$ and $\sigma$.

Set of All Permutations
The set of all permutations of $n$ letters is denoted $S_n$.

Cycle Notation
The two-row notation is a cumbersome way of defining a permutation.

Instead, the cycle notation is usually used instead.

Also denoted as
Some sources use $S \left({n}\right)$ for $S_n$.

Some sources use round brackets for the cycle notation: $\begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 4 \end{pmatrix}$

Also see

 * Symmetric Group