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 \\ \pi \paren 1 & \pi \paren 2 & \pi \paren 3 & \ldots & \pi \paren 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 denoted as
Some sources use square brackets for the two-row notation:


 * $\pi = \begin{bmatrix}

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

Also see

 * Definition:Cycle Notation