Two-line notation for permutations

Definition
The two-line notation is a notation used to describe a permutation on a (usually finite) set.

For a finite set
Suppose $$S$$ is a finite set and $$\sigma:S \to S$$ is a permutation. The two-line notation for $$\sigma$$ is a description of $$\sigma$$ in two aligned rows.

The top row lists the elements of $$S$$, and the bottom row lists, under each element of $$S$$, its image under $$\sigma$$.

If $$S = \{ a_1, a_2, \dots, a_n \}$$, the two-line notation for $$\sigma$$ is:

$$\begin{pmatrix}a_1 & a_2 & \dots & a_n \\ \sigma(a_1) & \sigma(a_2) & \dots & \sigma(a_n)\end{pmatrix}$$.

The two-line notation for a permutation is not unique. Given a different enumeration for the set $$S$$, both rows change accordingly.

If the enumeration of the elements of $$S$$ is fixed once and for all, the top line can be dropped, giving rise to the one-line notation for permutations.

For a countably infinite set
For a countably infinite set, we can use the two-line notation, with both lines being infinitely long.

Examples of the two-line notation for finite sets
Let $$S =\{ 1,2,3,4 \}$$ and $$\sigma$$ be defined as $$\sigma(1) = 2$$, $$\sigma(2) = 4$$, $$\sigma(4) = 3$$, and $$\sigma(3) = 1$$. The two-line notation for $$\sigma$$ is:

$$\begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 4 & 1 & 3 \\ \end{pmatrix}$$.

Examples of the two-line notation for infinite sets
Consider $$S$$ to be the set of all integers and $$\sigma$$ as the map $$x \mapsto x + 1$$. Then, the two-line notation for $$\sigma$$ is:

$$\begin{pmatrix}\dots & -2 & -1 & 0 & 1 & 2 & \dots \\ \dots & -1 & 0 & 1 & 2 & 3 & \dots \end{pmatrix}$$.