The Burnside Lemma looks like it should have an intuitive explanation. Does anyone have one?
As an example, we consider the number of ways of colouring a cube with n colours with uniqueness up to rotation. We call each unique colouring where rotations are not allowed a static colouring and each unique one where they are allowed a dynamic colouring. We define the the set of orbits to be the (disjoint) static colourings that correspond to each dynamic colouring. We will use rotations to mean a rotation that makes the cube occupy the same space, and as being unique if it is a unique function from the cube to the cube. This includes the identity rotation. Intuitively, the lemma says:
Proposition 1. #Orbits * #Rotations = Sum for each rotation r of #static colourings unchanged by this rotation
We will now consider each orbit O separately. Pick a static colouring c inside O. Suppose two (possibly equal) rotations p, q give the same static colouring, d, when applied on c. Then p−1q fixes d. Additionally, suppose r (possibly the identity) fixes d. p−1pr will also fix d. So pr will take c to d. Since pr is different for each r, and p−1q is different for each q, the mapping functions are injective in both directions and there is a bijection between the q and r values.
So, for each O, the number of rotations is the sum over each static colorings x in O times the number of rotations producing x. This can be rewritten as the sum over each rotation r of the number of static colourings in O fixed by r (due to the bijection in the previous paragraph). We get proposition 1 by adding over all O.
The general proof is quite similar to this, except that it uses group theory.