Question

Using Burnside's Lemma, determine a formula for the number of orbits under the action of D8 on the set of colourings.

(We colour each side of a square with one of k ≥ 1 colours.)

Answer 1

let's call the set of colored triangles X. The number of elements in X equals .

Burnside's lemma gives us:

The number of orbits

  ,

where Xg is the set of elements that are fixed by .

e fixes all the elements of X, so .

r also fixes all the elements of X because a rotation doesn't really produce a different triangle.

The same goes for .

s fixes the triangles that have at least two edges with the same color, so . And since rotations don't really change anything, the same goes for and .

So now we have

and so the number of orbits is