Weighted Burnside lemma
= Weighted Burnside lemma
For an orbit-invariant weight $w$ on a finite <group action> $G\curvearrowright X$, the sum of one weight per orbit is $|G|^{-1}\sum_g\sum_{x:gx=x}w(x)$. Double counting gives $\sum_x|G_x|w(x)$ before dividing by $|G|$; each orbit contributes $|G|$ copies of its common weight. Thus the usual <Burnside's lemma> extends to weights in any commutative coefficient ring.