Definition:Permutation on n Letters/Cycle Notation

Definition
Let $\N^*_k$ be the subset 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 Cycle Decomposition, all permutations can be defined as the product of disjoint cycles, and it doesn't matter in what order as Disjoint Permutations Commute.

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.

Canonical Representation
The permutation:


 * $\begin{bmatrix}

1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 4 & 3 & 5 \end{bmatrix}$

can be expressed in cycle notation as:


 * $\begin{bmatrix} 1 & 2 \end{bmatrix} \begin{bmatrix} 3 & 4 \end{bmatrix}$

or as:


 * $\begin{bmatrix} 3 & 4 \end{bmatrix} \begin{bmatrix} 5 \end{bmatrix} \begin{bmatrix} 1 & 2 \end{bmatrix}$

or as:


 * $\begin{bmatrix} 4 & 3 \end{bmatrix} \begin{bmatrix} 2 & 1 \end{bmatrix}$

etc.

However, only the first is conventional. This is known as the canonical representation.

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