Enter the number of elements in the set, each divisor of the number of elements, the Euler’s totient function of each divisor, and the number of cycles of length of each divisor into the calculator to determine the cycle index.

Cycle Index Formula

The following formula is used to calculate the cycle index. 

Z(Sn) = 1/n * Σ (d|n) φ(d) * a^{n/d}

Variables:

  • Z(Sn) is the cycle index n is the number of elements in the set
  • d is a divisor of n
  • φ(d) is the Euler’s totient function, which counts the positive integers up to a given integer d that are relatively prime to d
  • a is the number of cycles of length d

To calculate the cycle index, divide 1 by the number of elements in the set. Then, sum the product of the Euler’s totient function of each divisor of the number of elements and the number of cycles of length of that divisor, raised to the power of the number of elements divided by that divisor. Multiply the first result by the sum to get the cycle index.

What is a Cycle Index?

A Cycle Index is a mathematical tool used in combinatorics, the study of counting and arranging objects. It is a polynomial that describes the number of ways an object can be arranged or transformed while maintaining its overall structure. This is particularly useful in problems involving symmetry, where certain arrangements are considered identical. The Cycle Index is derived from the symmetry group of the object, with each term in the polynomial representing a different type of symmetry transformation.

How to Calculate Cycle Index?

The following steps outline how to calculate the Cycle Index using the given formula:

  1. First, determine the number of elements in the set (n).
  2. Next, determine the divisors of n (d).
  3. Next, calculate Euler’s totient function for each divisor (φ(d)).
  4. Next, determine the number of cycles of length d (a).
  5. Next, substitute the values into the formula: Z(Sn) = 1/n * Σ (d|n) φ(d) * a^n/d.
  6. Finally, calculate the Cycle Index.
  7. 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 elements in the set (n) = 6

Divisors of n (d) = 1, 2, 3, 6

Euler’s totient function (φ(d)) = 1, 1, 2, 2

Number of cycles of length d (a) = 1, 3, 2, 1