Definition:Permutation on n Letters/Two-Row Notation

Definition
Let $\pi$ be a permutation on $n$ letters.

The two-row notation for $\pi$ is written as two rows of elements of $\N_n$, as follows:


 * $\pi = \begin{pmatrix}

1 & 2 & 3 & \ldots & n \\ \map \pi 1 & \map \pi 2 & \map \pi 3 & \ldots & \map \pi n \end{pmatrix}$

The bottom row contains the effect of $\pi$ on the corresponding entries in the top row.

Also defined as
Some sources use two-row notation to define mappings which are not necessarily permutations. However, this is rarely done because of its general unwieldiness.

Also known as
Some sources refer to a presentation in two-row notation as a tableau.

Some sources use square brackets for the two-row notation:


 * $\pi = \begin{bmatrix}

1 & 2 & 3 & \ldots & n \\ \map \pi 1 & \map \pi 2 & \map \pi 3 & \ldots & \map \pi n \end{bmatrix}$

Also see

 * Definition:Cycle Notation