Cycle index for necklace with 12 beads

Question

I was trying to search for ways to calculate cycle index of a necklace with 12 beads and stumbled upon the formula that gave the answer as $\frac{{x_1}^{12} + {7x_2}^6 + {2x_3}^4 + {2x_4}^3 + {2x_6}^2 + {4x_{12}} + 6{x_1}^2{x_2}^5}{24}$. Is there any way to derive this formula using polya counting that has 12 rotations and 12 reflections?

Answer 1

@vk1234 Have you tried this way to get this formula? Let D be the set of 12 beads and R be the set of 12 postions namely 1,2,...,12. We use Polya's generalization theorem. We want the permutation group G consisting of only identity permutation as 12 beads are not interchangeable and the permutation group H consisting of 12 rotations and 12 reflections as you mentioned.

Finding cycle structure representation for each of these permutations we can get the cycle index for H as the one you wrote here.

For example,the cycle structure representation for identity permutation is $x_1^{12}$, for a reflection is $x_2^{6}$, for rotation of one position clockwise is $x_{12}^{1}$, for a rotation of one position clockwise followed by a reflection is $x_1^{2} x_2^{5}$, and so on.

I hope you have got the answer, the number of distinct necklaces as $\frac{(12^{12} + 6\times 2^{12})}{24}$ by using Generalization of Polya's Theorem.