Definition:Permutation on n Letters

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

A permutation of $$n$$ letters is a permutation $$\pi: \mathbb{N}^*_n \to \mathbb{N}^*_n$$.

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 $$S_n$$.

Two-Row Notation
Let $$\pi$$ be a permutation on $$n$$ letters.

The two-row notation for $$\pi$$ is written as two rows of elements of $$\mathbb{N}^*_n$$, as follows:

$$\pi = \begin{bmatrix} 1 & 2 & 3 & \ldots & n \\ \pi \left({1}\right) & \pi \left({2}\right) & \pi \left({3}\right) & \ldots & \pi \left({n}\right) \end{bmatrix} $$

The bottom row contains the effect of $$\pi$$ on the corresponding entries in the top row.

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

Instead, the cycle notation is usually used instead.

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.

Alternative Notation
Some sources use $$S \left({n}\right)$$ for $$S_n$$.

Some sources use round brackets for the two-row notation:

$$\pi = \begin{pmatrix} 1 & 2 & 3 & \ldots & n \\ \pi \left({1}\right) & \pi \left({2}\right) & \pi \left({3}\right) & \ldots & \pi \left({n}\right) \end{pmatrix} $$

... and the cycle notation: $$\begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 4 \end{pmatrix}$$