Orbit count is the average fixed-point count.Suggest a slogan
Let a finite group \(G\) act on a finite set \(X\). Let \(X/G\) be the set of orbits, and for \(g\in G\) let \(X^g=\theset{x\in X\suchthat g\cdot x=x}\) be the set of fixed points of \(g\). Then \[ \abs{G}\,\abs{X/G}=\sum_{g\in G}\abs{X^g}, \] so the number of orbits is the average number of fixed points of an element of \(G\).
Count the set \(\theset{(g,x)\in G\times X\suchthat g\cdot x=x}\) in two ways. Summing over \(g\) gives \(\sum_{g\in G}\abs{X^g}\). Summing over \(x\) gives \(\sum_{x\in X}\abs{G_x}=\sum_{x\in X}\abs G/\abs{G\cdot x}\) by the orbit-stabilizer theorem, and the \(\abs{G\cdot x}\) points of each orbit contribute \(\abs G\) in total, so this sum is \(\abs G\,\abs{X/G}\).
State Burnside's orbit-counting formula.