Editing Permutation (section)

===One-line notation===
If there is a "natural" order for the elements of ''S'',{{efn|The order is often implicitly understood. A set of integers is naturally written from smallest to largest; a set of letters is written in lexicographic order. For other sets, a natural order needs to be specified explicitly.}} say <math>x_1, x_2, \ldots, x_n</math>, then one uses this for the first row of the two-line notation:
: <math>\sigma = \begin{pmatrix}
  x_1         & x_2         & x_3         & \cdots & x_n \\
  \sigma(x_1) & \sigma(x_2) & \sigma(x_3) & \cdots & \sigma(x_n)
\end{pmatrix}.</math>

Under this assumption, one may omit the first row and write the permutation in ''one-line notation'' as
: <math>\sigma = \sigma(x_1) \; \sigma(x_2) \; \sigma(x_3) \; \cdots \; \sigma(x_n) </math>,
that is, as an ordered arrangement of the elements of ''S''.<ref>{{harvnb|Bogart|1990|p=17}}</ref><ref>{{harvnb|Gerstein|1987|p=217}}</ref> Care must be taken to distinguish one-line notation from the cycle notation described below: a common usage is to omit parentheses or other enclosing marks for one-line notation, while using parentheses for cycle notation. The one-line notation is also called the ''[[Word (mathematics)|word]] representation''.<ref name="Aigner2007"/>

The example above would then be:<blockquote><math>\sigma = \begin{pmatrix}
  1 & 2 & 3 & 4 & 5 & 6 \\
  2 & 6 & 5 & 4 & 3 & 1
\end{pmatrix} 
= 2  6  5  4  3  1.</math> </blockquote>(It is typical to use commas to separate these entries only if some have two or more digits.)

This compact form is common in elementary [[combinatorics]] and [[computer science]]. It is especially useful in applications where the permutations are to be compared as [[Partially ordered set|larger or smaller]] using [[lexicographic order]].