Burnside's Lemma states that, given a set $X$ acted on by a group $G$, 


$$|X/G|=\frac{1}{|G|}\sum\_{g\in G}|X^g|$$


where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of fixed points of $g$. In other words, the number of orbits is equal to the average number of fixed points of an element of $G$.


Is there any way in which the fixed points of an element $g$ can be thought of as orbits? I had wondered aloud on my recent question [here](https://mathoverflow.net/questions/50025/problems-where-we-cant-make-a-canonical-choice-solved-by-looking-at-all-choices) how (or if) Burnside's Lemma can be interpreted as having the same kind of object on both sides, so as to be a "true" average theorem, e.g.


"number of orbits = average over $g\in G$ of (number of orbits satisfying (something to do with $g$))"


or 


"number of orbits = average over $g\in G$ of (number of orbits of some new action which depends on $g$)"


Since Qiaochu stated the comments to my question that he suspects Burnside's Lemma can be categorified, and that this may be related, I have also added that tag.



Intuitive explanation of Burnside's Lemma
I'm not sure I'd call this a categorification, but the way I think of Burnside's Lemma is as follows.


Consider the subset $Z \subset G \times X$ consisting of pairs $(g,x)$ such that $g\cdot x =x$, where by $\cdot$ I just mean the action of $G$ on $X$.


The cartesian product $G \times X$ comes with the two surjections $\pi\_G : G \times X \to G$ and $\pi\_X : G \times X \to X$, and you can compute the cardinality of $Z$ either along the fibres of $\pi\_G$ or along the fibres of $\pi\_X$: the former gives you the sum over the fixed point sets, whereas the latter gives you a sum over the stabilizers. Then the orbit-stabilizer theorem does the rest.


Thanks to @Arrow who pointed out the link in my comment was broken. Here's hopefully a link that works to [the same one-page document](https://www.dropbox.com/s/fq9j7jqywxi6wrj/Polya.pdf?dl=0).


