Burnside's Lemma.

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Gnanf

Mathematics

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Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible.

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1. Find the number of nonequivalent 3-colorings of the sides of a hexagon, where two colorings are considered equivalent if one of them can be rotated onto the other by any of the 6 rotations by multiples of 60 degrees. 2. Find the number of nonequivalent 2-colorings of the 16 squares of a 4 x 4 checkerboard, where two colorings are equivalent if one of them can be rotated onto the other by any of the 4 rotations of the checkerboard by multiples of 90 degrees. 3. Find the number of ways to distribute 6 indistinguishable balls into 3 indistinguishable boxes. Hint: Start by first pretending that the 3 boxes are distinguishable, say with names A, B, C. Now consider the symmetric group on the 3 boxes A, B, C, consisting of the 6 permutations {id = (A)(B)(C), 71 = (AB)(C), 712 = (AC)(B), 13 = (BC)(A), 14 = (ABC), 75 (BAC)}. Observe how the distributions where the boxes are indistinguishable are in one to one correspondence with the orbits of distributions under this group when the boxes are distinguishable. Now use Burnside's Lemma to find the number of such orbits. 4. Find the number of 2-colorings of the vertices of the binary tree T of depth 2, where two colorings are equivalent if one be transformed to the other by some automorphism of T. (I will discuss automorphisms of graphs in class.)
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Explanation & Answer

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Problem 1:
The set X on which G  D 6 acts is the set of functions from the set of sides of the hexagon to the set of
3 colors {color 1, color 2, color 3}. Hence, we have the following table of fixed point sets

By Burnside’s lemma, the number ( N ) of nonequivalent 3-colorings of the sides of a hexagon is
calculated by

1 6
3  3  34  4  33  2  32  2  31 

1...

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