Two-line notation for permutations

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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}.