Definition:Permutation on n Letters/Cycle Notation

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

Let $\rho: \N^*_n \to \N^*_n$ be a permutation of $n$ letters.

The $k$-cycle $\rho$ is denoted $\begin{bmatrix} i & \rho \left({i}\right) & \ldots & \rho^{k-1} \left({i}\right) \end{bmatrix}$.

From Existence and Uniqueness of Cycle Decomposition, all permutations can be defined as the product of disjoint cycles.

As Disjoint Permutations Commute, the order in which they are performed does not matter.

So, for a given permutation $\rho$, the cycle notation for $\rho$ consists of all the disjoint cycles into which $\rho$ can be decomposed, concatenated as a product.

It is conventional to omit 1-cycles from the expression, and to write those cycles with lowest starting number first.

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