Two-line notation for permutations: Difference between revisions

Latest revision as of 21:28, 21 January 2010

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 $σ : S \to S$ is a permutation. The two-line notation for $σ$ is a description of $σ$ 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 $σ$ .

If $S = {a_{1}, a_{2}, \dots, a_{n}}$ , the two-line notation for $σ$ is:

$(\begin{matrix} a_{1} & a_{2} & \dots & a_{n} \\ σ (a_{1}) & σ (a_{2}) & \dots & σ (a_{n}) \end{matrix})$ .

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 $σ$ be defined as $σ (1) = 2$ , $σ (2) = 4$ , $σ (4) = 3$ , and $σ (3) = 1$ . The two-line notation for $σ$ is:

$(\begin{matrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 1 & 3 \end{matrix})$ .

Examples of the two-line notation for infinite sets

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

$(\begin{matrix} \dots & - 2 & - 1 & 0 & 1 & 2 & \dots \\ \dots & - 1 & 0 & 1 & 2 & 3 & \dots \end{matrix})$ .

Revision as of 19:22, 16 March 2009 (view source) Vipul (talk \| contribs) (New page: ==Definition== The '''two-line notation''' is a notation used to describe a permutation on a (usually finite) set. ===For a finite set=== Suppose <math>S</math> is a finite set and <mat...)		Latest revision as of 21:28, 21 January 2010 (view source) Vipul (talk \| contribs) (→‎Examples)
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