In how many ways can you color the edges of the triangle with 2 color (e.g. red and blue)?
- There are $m^3$ triangles fixed by the identity (1)(2)(3)
- There are $m^2$ triangles fixed under reflection, and there are 3 reflections (1)(23), (2)(13), (3)(12)
- There are $m^1$ triangles fixed under rotation, and there are 2 rotation(123), (132)
I know how many ways there are with rotation and reflections (2 color):
$\cfrac{1}{6} (2\cdot2^1 + 3\cdot2^2 + 1\cdot2^3) = 4$
I know how many ways there are with only rotation:
$\cfrac{1}{3} (2\cdot2^1 + 0\cdot2^2 + 1\cdot2^3) = 4$
I don't know how many ways there are with only reflections:
$\cfrac{1}{4} (0\cdot2^1 + 3\cdot2^2 + 1\cdot2^3) = 5$
I don't understand this last example. Is it correct? can anyone paint or describe it to me?