 Weighted Burnside lemma

Let a finite group $G$ act on a finite set $X$, with an orbit-invariant weight $w:X\to R$ in a commutative ring. Double counting gives the division-free identityIndeed, each $x$ is counted $|G_x|$ times, and each orbit contributes $|O||G_x|=|G|$ copies of its common weight. If $|G|$ is invertible in $R$, dividing gives the weighted average formula of Burnside's lemma. For formal monomial weights one can first count over the integers and then specialize into any commutative ring; this avoids dividing by zero in modular characteristic.