Find the number of different types of circular necklaces that could be made from the sets of beads
- 7 black and 5 white beads
We need to solve this by the Polya-Burnside method of enumeration:
Since we have 7 black and 5 white beads, it means we have a regular 12-gon, and we need to list all the permutations that correspond to a particular symmetry.
A rotation by $\pi/6 \rightarrow (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)$ permutation, which is $p_{1}^{6}$
A rotation by $\pi/3 \rightarrow (1, 3, 5, 7, 9, 11)(2, 4, 6, 8, 10, 12)$ permutation, which is $p_{6}^{2}$
...
From this we will be able to get the cycle index polynomial of $D_{12}$, and this will ultimately allow us to find the number of different types of circular neckalaces
The problem that I'm encountering is just corresponding each symmetry to a permutation to get the cycle index polynomial $D_{n}$, where $n \geqslant 7$. This is just extremely painful to do. In our case, we have to correspond 24 symmetries! This is just too much, and it's very easy to make a mistake.
My question: Is there some easier way to get the cycle index polynomial of $D_{n}$ without corresponding each symmetry to a permutation?