Orbit count is the average fixed-point count.Suggest a slogan
Let \(G\) be a finite group acting on a finite set \(X\). Let \(X/G\) be the set of orbits, and for \(g\in G\) let \(\Fix(g) = \theset{x\in X \st gx=x}\). Then \[ \# (X/G) = \frac{1}{\# G}\sum_{g\in G} \# \Fix(g). \]
The number of orbits is the average over \(G\) of the number of fixed points.