A cyclic permutation is built from one or more sets of elements in cyclic order. The notion cyclic permutation is used in different, but similar ways:
Contents |
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 + t − k] for i = k − t + 1, k − t + 2, ..., k.
Note: Every cyclic permutation of definition type 1 will be constructed with exactly gcd (k, t) disjoint cycles; see cycles and fixed points. Cyclic permutations of definition type 1 are also called rotations. Example:
- <math>
\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 7 & 6 & 8 \\ 3 & 4 & 5 & 7 & 6 & 8 & 1 & 2 \end{pmatrix}</math> 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:
- <math>
\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 7 & 6 & 8 \\ 4 & 5 & 7 & 6 & 8 & 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 4 & 6 & 2 & 5 & 8 & 3 & 7 \\ 4 & 6 & 2 & 5 & 8 & 3 & 7 & 1 \end{pmatrix}</math>
Definition 3
A permutation is called a cyclic permutation if and only if only 1 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:
- <math>
\begin{pmatrix} 1 & 4 & 6 & 8 & 3 & 7 & 2 & 5 \\ 4 & 6 & 8 & 3 & 7 & 1 & 2 & 5 \end{pmatrix}</math>
