Permutations

[mathsishardd]

I'm trying to find the permutations of necklace with 4/6/8 beads each group of different color, each bead in a group indistinguishable from the rest.

All I have so far is (18!/(4!*6!*8!)+9!/2!*3!*4!+??)/36 but I am unsure of what I need to replace the ?? with.

I've seen that it's sometimes replaced with (n/2)! and sometimes (n/2)*(n/2)! + (n/2)*(n/2)*2 and it's sometimes completely different but unable to get the logic.

Looking for tips

 

[Cubist]

I'm trying to find the permutations of necklace with 4/6/8 beads each group of different color, each bead in a group indistinguishable from the rest.
All I have so far is (18!/(4!*6!*8!)+9!/2!*3!*4!+??)/36 but I am unsure of what I need to replace the ?? with.

So each necklace will be made from 18 beads. The 18 beads have the following colours (for example) 4 are green, 6 white and 8 are blue. If this is true then there are 18!/(4!*6!*8!) ways to make the necklace. These are the permutations of a multiset.

 

[mathsishardd]

Since this is a necklace, rotating/flipping won't give a different permutation, this is to do with Polya Enumeration but I can't seem to find how to solve it.

 

[Cubist]

I don't understand Polya Enumeration. But I worked out an answer, using a computer program, since this might be helpful in your search for the correct method:- With flipping and rotations, I found 255920 permutations. With rotations only 510580.

NOTE: I am concerned, however, that my code disagrees with this wikipedia page on some answers. For example, in the upper right image, the case with 5 beads 1/2/2, ...



They show 3 permutations, but I think the "missing" one shown is also valid. Do you agree?