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 < c < ... < c) and the mapping of P may be written as:
- p(c ) = c for i = 1, 2, ..., k − t, and
- p(c) = c 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 of equal length; see cycles and fixed points.
Cyclic permutations of definition type 1 are also called rotations, or circular shifts.
Example:
is a cyclic permutation with offset 2. It may be constructed with gcd(8, 2) = 2 cycles; see image. The used order is: c := 7, c :=6, c = i else.
Read more about this topic: Cyclic Permutation
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