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

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:

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