Circular Permutation

The term “Permutation” is known for the arrangement of entities in a specific order. It is a special tool in Math that arranges objects in a sequence of linear patterns. The tool is widely used in the calculation process of multiple objects and to arrange them in linear orders. The tool helps in finding out the probable outcomes in which objects can be arranged. It is used when there is a need for an ordered sequence of objects. The use of combinations is related to the finding of specific groups. “Circular permutation” deals with the arrangement of objects around a fixed circle.

What is Circular Permutation?

The tool “Circular Permutation” is the calculation of the total number of procedures that can be adopted in order to place distinct objects perfectly around a fixed circle. It is done by following two processes to be precise. The first one is done by “Clockwise and Anticlockwise orders are different” and the second one is done by “Clockwise and Anticlockwise orders are the same”.

The process of cyclic permutation follows 2 formulas 

1. Pn  = (n-1)!
2. Pn= (n-1)!/2!

The formula implies that cyclic permutation is the permutation of the elements of some set (X) with the mapping of the subset [S] of (X) to each other in a cyclic fashion. It also fixes all the elements o (X). In case the subset [S] has K number of elements then it is known as k-cycle. 

Characteristics of Circular Permutation

The characteristics of circular permutation in the context of mathematical calculation can be explained as follows:

• One element is always fixed in the process of circular permutation and the other element is variable and is arranged in retain to the fixed element.
• There is no start or end to the process because it is cyclic in nature. The arrangement of elements is done in a looped process that does not have an end until acted upon.
• Marking the start and end of the circle is not possible as it functions in a loop and arrangement happens randomly in a linear pattern.

Application of cyclic permutation

• It is extensively used in microbiology by protein engineers that determine the substitution of amino acids to alter the functions and properties of the bio-macro molecules. 
• It helps in the process of re-organizing the polypeptides present in the protein chain synthesis
• It helps in deforming the protein structure and enables the identification of multiple connectivity nodes in the process.
• A circular permutation is crucial for the framing of various seating arrangements.

Circular Permutation Arrangement

According to the properties of cyclic permutation, it can be said that four persons A, B, C, D can arrange themselves in four different ways. Since there is no start or end to the loop of circular permutation, one of the elements is always fixed and the other is position independent. The free element is structured according to the value of the fixed element. The number of ways in which the other person can arrange themselves, can keep in mind that the position of A is fixed. (4-1)!= 3!= 6

Hence it proves that while keeping a fixed node in the circular permutation the other three people can arrange themselves in 6 possible ways.

Transposition

“Transposition” is a unique part of the cyclic permutation. It consists of a cycle of only two elements. A permutation is expressed as the value of the composition (product) of transpositions that formally form the generators for the group. On the calculation of a value of n integers that result can be termed as the product formed by the multiplication of the adjacent sets of data. The opposite is also possible which involves the decomposition of a permutation into two parts known as the singular adjacent transpositions. These can be expressed in terms of disjoint cycles and can be split by the division of the cycle by the cycle length provided. The initial request for the calculation of transpositions can be termed as the movement of elements to alternate positions in the cycle. One of the main results on the application of symmetric groups states that all of the decompositions according to the given permutation into transpositions have quite a number of transpositions, or they belong to the odd number of transpositions. The prospect highly describes that parity in a permutation is a concept that is quite heavily used in the application of circular permutation.

Notation

On Consideration, to have n number of items from which an arrangement needs to be made for r of them. The number of ways in which the arrangement can take place is given by permutation as nPr = n!(n−1)!n! ⁄ (n – r)!, 0 ≤ r ≤ n.

It is always considered in the calculator process that the factorial of 0 is 1.

Case 1

The first choice is the pencil

Circular Permutation

Case 2

The first choice is the pen

Circular Permutation

In both cases, the number of ways is 6 = 2 × 3 or 3 × 2. Here, 6 = 3!

In factorial notation, n! = n × (n – 1) × (n – 2) × … × 2 × 1.

Conclusion

It can be concluded that circular permutation is a very important part of the mathematical tools used in microbiological processes and framings. The circular permutation is unique in nature because it does not have any beginning or end and implements the placing of objecting in a circular pattern. The arrangement is linear in format and is mostly used in the synthesis of protein particles. It is also used to determine the seating positions in a circular manner. A circular permutation is formulations and theories that are simple to understand and useful to implement.