Orientation-Less Circular Permutation with In-distinguishable Objects

Let's say I have 4/6/8 objects placed in a circle, each group of its specific color, indistinguishable from one another in each group.

Finding the possible arrangements with no specific starting point isn't that hard - find all of the possible combinations as if they were in a line and divide by the amount of objects (510,510).

Now though, I need it so that the orientation doesn't matter - reverse it and you'd get the same thing (example: 4321 = 1234).

I thought, firstly, just divide by 2 - easy!

But then I realized that the reversed combination could equal the non-reversed so dividing by 2 could provide not all the different combinations but less (checked it with a program and proved true).

How do I go from here?

• When you did the first part with just rotations, did you take into account that e.g. aabbbccccaabbbcccc is unchanged by rotation by $9$ (I.e. half a circle) ? So you can’t just divide by $18$ because some linear assignments don’t give $18$ different assignments under the $18$ rotations. The thing to look up is Burnside’s Lemma for both this issue and what happens with reflections.
– Ned
May 10, 2020 at 12:14
• @Ned No, I didn't :( Thanks, I will look into that May 10, 2020 at 12:25