Understanding Cyclic Permutations

A cyclic permutation is a specific type of permutation where the elements of a set are shifted in a circular manner. Each element moves to the position of the next element in sequence, and the last element moves to the first position. This concept is widely applicable in various fields including combinatorics, group theory, and computer science. In this article, we will delve into the properties, notations, and applications of cyclic permutations.

What is a Cyclic Permutation?

Consider a set of elements, for example: {1 2 3 4}. A cyclic permutation of this set could be:

Original: 1 2 3 4 Cyclic permutation: 4 1 2 3

In this case, each element moves to the position of the next element, and the last element (4) moves to the front. This shifting in a circular manner is the essence of a cyclic permutation.

Properties of Cyclic Permutations

Length

A cyclic permutation of n elements can be represented as a single cycle of length n. This means that the entire set of elements is treated as a single continuous loop.

Notations

Cyclic permutations are often denoted in cycle notation. For instance, the cyclic permutation of {1 2 3 4} can be written as:

1 2 3 4

Number of Cyclic Permutations

The number of distinct cyclic permutations of n elements is (n-1)!. This is because, by fixing one element, the remaining n-1 elements can be arranged in (n-1)! distinct ways.

Applications of Cyclic Permutations

Cyclic permutations have a wide range of applications, including combinatorics, group theory, and various fields of mathematics. They are also useful in computer science, particularly when dealing with circular data structures or arrangements. For instance, in a computer program, cyclic permutations might be used to rotate elements in a circular buffer or to organize elements in a circular fashion.

Examples and Further Details

A cyclic permutation or cycle is a permutation of the elements of some set X, which maps the elements of a subset S of X to each other in a cyclic fashion while fixing all other elements of X. The number of such cycles can vary based on the size of the subset S. For example, for n3:

Linear Permutations

The number of linear (standard) permutations of n items is given by n!. For n3, we have:

3! 123 6

Permutations: abc, acb, bac, bca, cab, cba

Cyclic Permutations

The number of cyclic permutations is given by (n-1)!. For n3, we have:

3–1! 2! 2

Cyclic permutations: abc, acb

Furthermore, a cyclic permutation or cycle is a of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion while fixing all other elements of X. If S has k elements, the cycle is called a k-cycle.

For example, given X {1 2 3 4}:

Permutation: 1 goes to 3, 3 goes to 2, 2 goes to 4, 4 goes to 1. This is a 4-cycle because all elements of S X (4 elements) are involved. Permutation: 1 goes to 3, 3 goes to 2, 2 goes to 1, 4 goes to 4. This is a 3-cycle because S {1 2 3} (3 elements) and 4 is fixed. Permutation: 1 goes to 3, 3 goes to 1, 2 goes to 4, 4 goes to 2. This is not a cyclic permutation because it separately permutes the pairs {1 3} and {2 4}.

For more detailed information, you can follow the link below to the Wikipedia page on cyclic permutations: Cyclic permutation - Wikipedia.