Enter the number of permutations and the cycle size into the calculator to determine the cycle index. The cycle index is a way to describe the symmetry of a combinatorial object.
Cycle Index Formula
The following formula is used to calculate the cycle index (Z).
Z = (1 / k) * n! / (n - k)!
Variables:
- Z is the cycle index
- n is the number of permutations
- k is the cycle size
To calculate the cycle index, divide the factorial of the number of permutations by the factorial of the difference between the number of permutations and the cycle size, then divide the result by the cycle size.
What is a Cycle Index?
A cycle index is a polynomial that encodes information about the permutation group of a combinatorial object and its symmetry. It is used in various fields such as group theory, combinatorics, and chemistry to study the structural properties of objects and their symmetrical features.
How to Calculate Cycle Index?
The following steps outline how to calculate the Cycle Index.
- First, determine the number of permutations (n).
- Next, determine the cycle size (k).
- Ensure that the cycle size (k) is less than or equal to the number of permutations (n).
- Next, gather the formula from above = Z = (1 / k) * n! / (n - k)!.
- Finally, calculate the Cycle Index (Z).
- After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem :
Use the following variables as an example problem to test your knowledge.
number of permutations (n) = 5
cycle size (k) = 3
