Orbit combinatorics

Orbit combinatorics is about counting objects that are considered the same under some symmetry. For example, when coloring the faces of a cube, two colorings that can be rotated to look the same are considered identical. The main tool is to group objects into 'orbits' under the action of a group (like rotations), and count the number of different orbits.

Important properties

• An orbit is a set of objects that can be transformed into each other by the symmetries of the problem.

• The number of distinct objects up to symmetry is the number of orbits.

• Burnside's Lemma is a key result: it says the number of orbits equals the average number of objects fixed by each symmetry.