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Find the number of different types of circular necklaces that could be made from the sets of beads

  1. 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?

RobPratt
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Amir Zhang
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    Do you need the polynomial just for the case of $5$ white and $7$ black beads? If so the question becomes quite easy – $5$ and $12$ are coprime, so straight division $\binom{12}5/12=66$ gives the answer. – Parcly Taxel Nov 06 '21 at 14:50
  • The Pólya enumeration theorem also doesn't quite apply when the colour counts are provided. All $12$-necklaces are necessarily $C_{12}$-aperiodic from the coprimality mentioned above; also your necklaces are normally called bracelets since only they allow flipping over and have $D_n$ symmetry. – Parcly Taxel Nov 06 '21 at 14:56
  • @ParclyTaxel We are considering two necklaces to be the same if one can be transformed into another via a rotation. I am asking about the cycle index polynomial that is used in the Polya-Burnside method of enumeration. For example, The cycle index polynomial of $D_{6} is 1/12[x_{6}^1+3x_{1}^2x_{2}^2+4x_{2}^3+2x_{3}^2+2x_{6}]$ – Amir Zhang Nov 06 '21 at 14:58
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    But do you also allow reflections? If not, then the relevant group is $C_n$ and not $D_n$; if yes, you should be using bracelet instead. There is an inconsistency. – Parcly Taxel Nov 06 '21 at 15:00
  • @ParclyTaxel Yes, we allow reflections. I thought counting the number of circular necklaces is just the same as counting the number of ways to color a regular 12-gon using 7 black and 5 white colors – Amir Zhang Nov 06 '21 at 15:05
  • Ah, the specific cycle polynomial for $D_{12}$ is given here, which is quite easily generalisable to arbitrary $n$. – Parcly Taxel Nov 06 '21 at 16:54
  • There is also a fact sheet on necklaces and bracelets at this MSE link. – Marko Riedel Nov 06 '21 at 18:17
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    I have never heard of this distinction between necklaces and bracelets. It does not seem particularly apt, because necklaces can be turned over. – Derek Holt Nov 07 '21 at 08:25

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