The number of link and cluster states: the core of the 2D q state Potts model

Due to Fortuin and Kastelyin the q state Potts model has a representation as a sum over random graphs, generalizing the Potts model to arbitrary q is based on this representation. A key element of the random cluster representation is the combinatorial factor ΓG(C, E), which is the number of ways to form C distinct clusters, consisting of totally E edges. We have devised a method to calculate ΓG(C, E) from Monte Carlo simulations