There are 26 2 30 ways to color the vertices of a regular

There are 2^6 - 2 = 30 ways to color the vertices of a regular hexagon with blue and red (using each color at least once). Consider two such colorings identical\" if one can be obtained from the other by a rotation. How many different colorings are there?

Solution

solution_:In a regular hexagon with blue and red ( using each color at least one)

The number of assignments of 2 colors to the vertices that are preserved by a group element is

2^Number of vertex orbits under

since each vertex orbit can be assigned any color, and every vertex in any orbit must be colored the same.

Inputting this into Burnside\'s Lemma gives the number of assignments of 2 colors (inequivalent under rotations and reflections) as

=1/12(2^6+2^1+2^2+2^3+2^2+2^1+2^3+2^4+2^3+2^4+2^3+2^4)=13.

Precisely two of these inequivalent assignments of 2 colors have all colours the same: when they\'re all blue, and when they\'re all Red. That leaves 11 inequivalent assignments of 2 colors to the vertices where both colors are used. Answer