# Two-line notation for permutations

## Contents

## Definition

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

### For a finite set

Suppose is a finite set and is a permutation. The **two-line notation** for is a description of in two aligned rows.

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

If , the two-line notation for is:

.

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

If the enumeration of the elements of 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 and be defined as , , , and . The two-line notation for is:

.

### Examples of the two-line notation for infinite sets

Consider to be the set of all integers and as the map . Then, the two-line notation for is:

.