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June 10th, 2010, 08:30 PM   #1
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Number of Necklace/Bracelets With Fixed Number of Beads

We can compute the total number of k-ary necklaces/bracelets of length n using the Euler totient function:

http://en.wikipedia.org/wiki/Necklace_% ... atorics%29

Are there formulas for the number of k-ary necklaces/bracelets when the number of beads for each color is fixed?

For example, the number of necklaces of length 5, made up of three colors (red, green, blue), with 2 red beads, 2 green beads, and 1 blue bead. etc.
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June 12th, 2010, 05:57 PM   #2
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Re: Number of Necklace/Bracelets With Fixed Number of Beads

Yes, we can. But I don't completely remember how (it's been 2 years since I last saw it!)

I believe there is some information on approaches to this sort of problem in Wilf's Generatingfunctionology.
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June 13th, 2010, 12:03 AM   #3
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Re: Number of Necklace/Bracelets With Fixed Number of Beads

This problem seems similar to 'How many ways can you colour the sides of a cube using 3 colours' and we used group theory, orbits and stabilizers.
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