Cyclic permutation

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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[edit]

mapping of permutation

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]

mapping of permutation

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)

See also[edit]

References[edit]

  1. ^ Carmichael, Robert D. (1956) [1937], Introduction to the theory of Groups of Finite Order, Dover, p. 7, ISBN 0-486-60300-8  Carmichael uses the terms "cyclic" and "circular permutation" as synonyms.
  2. ^ Bogart, Kenneth P. (1990), Introductory Combinatorics (2nd ed.), Harcourt, Brace, Jovanovich, p. 486, ISBN 0-15-541576-X 
  3. ^ Gross, Jonathan L. (2008), Combinatorial Methods with Computer Applications, Chapman & Hall/CRC, p. 29, ISBN 978-1-58488-743-0