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The notion "cyclic permutation" is used in different, but related ways:
- the elements of S may be ordered (c < c < ... < 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.
Cyclic permutations of definition type 1 are also called rotations, or circular shifts.
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] = i else.
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
A permutation is called a cyclic permutation if and only if only one of the constructing cycles will have length > 1.
Every cyclic permutation of definition type 2 may be seen ″as a cyclic permutation of definition type 3 with zero fixed points.