TITLE: Nice application of the Cauchy?-Frobenius?-Burnside?-Pólya? formula
QUESTION [16 upvotes]: Burnside's Lemma, whose list of names is longer than the proof, says that the number of orbits of a permutation group is the average number of fixed points of its elements. It's a very elegant result, but I'm a bit disappointed by the fact that the examples given in the textbooks always amount to counting some colorings of a symmetric object, up to symmetry (the less original example probably being the cube). My question is then: do you know some funnier (but still rather direct) applications of this result?

REPLY [2 votes]: You can use it to count the number of isomorphism classes of representations of a quiver over a finite field; Burnside's lemma was used for this  purpose by Kac and Stanley (see Root Systems, Representations of Quivers and Invariant Theory by Victor G. Kac).