A cyclic permutation or circular permutation is a permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

 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

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

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

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

the elements of S may be ordered

Total order

In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...

 (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

Greatest common divisor

In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

 (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[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:

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

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

.

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

Example:

See also

• Cyclic permutation of integer
• Cycle notation
  Cycle notation
  In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles. This is also called circular notation and the permutation called a cyclic or circular permutation....
  
  
• Cycles and fixed points
• Stirling number
  Stirling number
  In mathematics, Stirling numbers arise in a variety of combinatorics problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second...
  
  
• Caesar cipher
  Caesar cipher
  In cryptography, a Caesar cipher, also known as a Caesar's cipher, the shift cipher, Caesar's code or Caesar shift, is one of the simplest and most widely known encryption techniques. It is a type of substitution cipher in which each letter in the plaintext is replaced by a letter some fixed number...
  
  
• Circular permutation in proteins
  Circular permutation in proteins
  Circular permutation is a process during evolution that changes the order of amino acids in a protein sequence, resulting in a protein structure with different connectivity, but overall similar three dimensional shape. As a consequence of the circular permutation, the N-terminus of one protein...