Editing Permutation (section)

===Two-line notation===
[[Augustin-Louis Cauchy|Cauchy]]'s ''two-line notation''<ref>{{cite journal |last1=Cauchy |first1=A. L. |title=Mémoire Sur le Nombre des Valeurs qu'une Fonction peut acquérir, lorsqu'on y permute de toutes les manières possibles les quantités qu'elle renferme |journal=Journal de l'École polytechnique |date=January 1815 |volume=10 |pages=1–28 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015074785612&seq=9 |trans-title=Memoir on the number of values which a function can acquire when one permutes within it, in all possible ways, the variables which it contains |language=French}}  See p. 4.
* [https://nonagon.org/ExLibris/cauchys-memoire-sur-le-nombre-des-valeurs English translation]</ref><ref>{{citation|title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory|first=Hans|last=Wussing|publisher=Courier Dover Publications|year=2007|isbn=9780486458687|page=94|url=https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA94|quote=Cauchy used his permutation notation—in which the arrangements are written one below the other and both are enclosed in parentheses—for the first time in 1815.}}</ref> lists the elements of ''S'' in the first row, and the image of each element below it in the second row. For example, the permutation of ''S'' = {1, 2, 3, 4, 5, 6} given by the function<blockquote><math>\sigma(1) = 2, \ \ \sigma(2) = 6, \ \ \sigma(3) = 5, \ \ \sigma(4) = 4, \ \ \sigma(5) = 3, \ \ \sigma(6) = 1   </math></blockquote>can be written as
: <math>\sigma = \begin{pmatrix}
  1 & 2 & 3 & 4 & 5 & 6 \\
  2 & 6 & 5 & 4 & 3 & 1
\end{pmatrix}.</math>
The elements of ''S'' may appear in any order in the first row, so this permutation could also be written:
: <math>\sigma = \begin{pmatrix}
  2 & 3 & 4 & 5 & 6 & 1 \\
  6 & 5 & 4 & 3 & 1 & 2
\end{pmatrix}
= \begin{pmatrix}
  6 & 5 & 4 & 3 & 2 & 1 \\
  1 & 3 & 4 & 5 & 6 & 2 
\end{pmatrix}.</math>