New derivation of the cluster cumulant formula

The cluster cumulant formula of Kubo is derived by appealing only to elementary properties of subsets and binomial coefficients. It is shown to be a binomial transform of the grand potential. Extensivity is proven without introducing cumulants. A combinatorial inversion is used to reformulate the expansion in the activity to one in occupation probabilities, which explicitly control the convergence. The classical virial expansion is recovered to third order as an example.