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June 10th, 2010, 08:30 PM  #1 
Newbie Joined: Oct 2009 Posts: 26 Thanks: 0  Number of Necklace/Bracelets With Fixed Number of Beads
We can compute the total number of kary 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 kary 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. 
June 12th, 2010, 05:57 PM  #2 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 2  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. 
June 13th, 2010, 12:03 AM  #3 
Newbie Joined: Jul 2009 From: England Posts: 25 Thanks: 0  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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