Nice application of the Cauchy?-Frobenius?-Burnside?-Pólya? formula 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?
 A: Yes, there are many more funny applications, for sure, here are three of them:


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*You can apply it in order to compute the sum of reciprocals of cardinals all all endomorphisms of a vector space of dimension $n$ over some field with $p$ elements ($p$ a prime).

*You can apply it in order to characterize those finite groups that have an abelian automorphism group.

*It can be applied in order to (pun intended!) calculate the multiplicative order of some
$u$ such that $u$ is co-prime to a given $n$ (that is, to calculate the smallest $k$ such that $n$ divides $u^k-1$).
I share with you the disappointment: this is such a general thing that it is only conceivable that the range of applications is bounded only by the imagination of the user...
A: 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).
A: Burnside's lemma can be used to prove the Polya enumeration theorem, which has many applications; see, for example, these two blog posts.  The application to the symmetric groups alone is the well-known exponential formula in combinatorics, which has many applications; see this blog post.
It also has applications to representation theory.  If $X$ is a set on which a group $G$ acts, then the free vector space on $X$ is a representation $V$ of $G$ with character $\text{Fix}(g)$.  Burnside's lemma and the orthogonality relations then tell you that the dimension of the invariant subspace of $V$ is the number of orbits of the action of $G$ on $X$.  They also tell you that if $V$ decomposes as a direct sum $\oplus n_k V_k$ where the $V_k$ are irreducible, then $\sum n_k^2$ is the number of orbits of $G$ acting on $X \times X$.  In particular, if $G$ acts double transitively there are two such orbits, so $V$ is the sum of a trivial representation and an irreducible representation.
(This application to representation theory, in turn, has applications to graph theory.  See this blog post.)
Edit:  Here are some MO and math.SE answers where I have used Burnside's lemma:


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*https://mathoverflow.net/questions/50253/can-this-nested-sum-be-expressed-in-terms-of-generalized-harmonic-numbers-and-the/50256#50256

*https://mathoverflow.net/questions/30112/elementary-combinatorial-identity-expressing-binomial-coefficients-as-an-alter/30114#30114

*Coloring the faces of a hypercube
I want to point out one of the applications I mention in one of the above blog posts which I think is particularly "funny": Fermat's little theorem!  Consider the cyclic group of order $p$ acting on the set of strings of length $p$ from an alphabet of size $a$.  By Burnside's lemma the total number of orbits is
$$\frac{1}{p} \left( a^p + (p-1)a \right)$$
since there is one element which fixes every string and $p-1$ elements which only fix strings which repeat one letter $p$ times.  The integrality of this number is equivalent to Fermat's little theorem.  (For a generalization, see these two blog posts.)
