# Cyclic permutation

A cyclic permutation[1] is a permutation in cyclic order of a set of elements.

The term circular permutation has the modern meaning of a permutation of objects arranged in a circle (without a distinguished starting object), but in older texts it was used as a synonym for cyclic permutation.

The notion of a "cyclic permutation" may be used in slightly different ways depending on whether the author does or does not want to include fixed points,

## Definition

A permutation is called a cyclic permutation if and only if it consists of a single nontrivial cycle (a cycle of length > 1).[2]

Example:

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 2 & 7 & 6 & 5 & 8 & 1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 4 & 6 & 8 & 3 & 7 & 2 & 5 \\ 4 & 6 & 8 & 3 & 7 & 1 & 2 & 5 \end{pmatrix} = (146837)(2)(5)$

Some authors restrict the definition to only those permutations which have precisely one cycle (that is, no fixed points allowed).[3]

Example:

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 5 & 7 & 6 & 8 & 2 & 1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 4 & 6 & 2 & 5 & 8 & 3 & 7 \\ 4 & 6 & 2 & 5 & 8 & 3 & 7 & 1 \end{pmatrix} = (14625837)$