Definition:Permutation on n Letters/Two-Row Notation

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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:

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


Examples

Permutations in $S_3$

The permutations on the symmetric group on $3$ letters $S_3$ can be depicted in two-row notation as:

$\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix} \qquad \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix} \qquad \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix}$
$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix} \qquad \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix} \qquad \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix}$


Permutation in $S_4$

The permutation on the symmetric group on $4$ letters $S_4$ defined as:

$1 \mapsto 3, 2 \mapsto 2, 3 \mapsto 4, 4 \mapsto 1$

can be depicted in two-row notation as:

$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 2 & 4 & 1 \end{pmatrix}$


Also see


Technical Note

The $\LaTeX$ code for \(\begin{pmatrix} a & b & c & d \\ d & b & c & a \end{pmatrix}\) is \begin{pmatrix} a & b & c & d \\ d & b & c & a \end{pmatrix} .


For a more compact presentation, the code:

\dbinom {a \ b \ c \ d} {d \ b \ c \ a}

can be used, which gives:

$\dbinom {a \ b \ c \ d} {d \ b \ c \ a}$

However, this technique does not work well when the width of the fields is non-uniform:

$\dbinom {m \ l \ l \ m} {m \ m \ l \ l}$


Sources

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