Hello everyone

I am trying to solve a coloring problem i.e find the number of distinct coloring's of a beaded necklace with colors. In doing so I want to derive an equation regarding the number of distinct orbits of a group G which are more applicable than:

In doing so I am considering the coloring of the vertices of a gon when two coloring's are regard the same if reachable by:

- The Cyclic Group
- The Dihedral Group

Acting on the gon.

So far I have figured out that for the Cyclic group the above (Burnside's Lemma) can be calculated easier as:

Or put in words the number of colors raised to the number of disjoint-cycles for each permutation. However when I consider the Dihedral Group I find that for the first elements, which are basically the group, I do the same, but for all the elements which can be expressed as an action for an rotation counterclockwise and a reflection, I need to add one to the number of disjoint cycles. As an example using and I find the two polynomials.

for this gives 130

Which makes sense as it fits the number of disjoint cycles, however I also find that:

for this gives 92

But for the elements , I find that the elements are , , , , and .

So my question is why do I have to add and extra to the number of disjoint cycles to elements .

Thank you