Cyclic permutation

A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.

The notion "cyclic permutation" is used in different, but related ways:

Definition 1

A permutation P over a set S with k elements is called a cyclic permutation with offset t if and only if

the elements of S may be ordered (c[1] < c[2] < ... < c[k]) and the mapping of P may be written as:
p(c[i] ) = c[i + t] for i = 1, 2, ..., k  − t, and
p(c[i]) = c[i + tk] for i = k − t + 1, k − t + 2, ..., k.

Note: Every cyclic permutation of definition type 1 will be constructed with exactly gcd (kt) disjoint cycles of equal length; see cycles and fixed points.

Cyclic permutations of definition type 1 are also called rotations, or circular shifts.

Example:

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & 7 & 6 & 1 & 8 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 7 & 6 & 8 \\ 3 & 4 & 5 & 7 & 6 & 8 & 1 & 2 \end{pmatrix} = (1356)(2478)$

is a cyclic permutation with offset 2. It may be constructed with gcd(8, 2) = 2 cycles; see image. The used order is: c[6] := 7, c[7] :=6, c[i] = i else.

Definition 2

A permutation is called a cyclic permutation if and only if it will be constructed with exactly 1 cycle.

Note: Every permutation over a set with k elements is a cyclic permutation of definition type 2 if and only if it is a cyclic permutation of definition type 1 with gcd(k, offset) = 1

Example:

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

Definition 3

A permutation is called a cyclic permutation if and only if only one of the constructing cycles will have length > 1.

Note: Every cyclic permutation of definition type 3 may be seen as an union of a cyclic permutation of definition type 2 and some fixed points.

Every cyclic permutation of definition type 2 may be seen ″as a cyclic permutation of definition type 3 with zero fixed points.

Example:

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