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Nice application of the Cauchy?-Frobenius?-Burnside?-Pólya? formula
Burnside's lemma states that the number of orbits in a group action is equal to the average number of fixed points the elements of the group have. It is well known for its many combinatorial uses, and this is exactly what I'm after - examples of such uses, hopefully as diverse as possible. The most famous example is counting the number of ways to color a cube, so I'm primarily interested in examples of a different nature.