Number of distinguishable ways to paint the sides regular n-gon using m colors

Asked Feb 28 '24 at 15:20

Active Feb 28 '24 at 15:20

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I'm trying to derive a general formula for the number of distinguishable ways to paint the sides regular $n-$gon using $m$ colors.

Two rules:

We don't have to use all of the $m$ colors and we have to account for rotations.

My progress:

We have to divide whatever we end up getting by $n$ to account for rotations. But from there I can't see a good next step. I know how to solve say, a square with 2 colors, but when I take the same approach for $n$ sides and $m$ colors, I get lost in recursion and inclusion-exclusion.

Any takers?

asked Feb 28 '24 at 15:20

MrMustache

3

https://en.wikipedia.org/wiki/Burnside's_lemma is the standard tool here. I think you will find the prime factorisation of $n$ matters a lot. – Arthur Feb 28 '24 at 15:24
3

This is the number of $m$-ary necklaces of length $n$. – joriki Feb 28 '24 at 15:27
thanks guys. got it – MrMustache Feb 28 '24 at 15:55

Number of distinguishable ways to paint the sides regular n-gon using m colors

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