Two-line notation for permutations
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 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: