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 $\Gamma_{\Graph{G}}(\Clusters,\Edges)$, which is the number of ways to form $\Clusters$ distinct clusters, consisting of totally $\Edges$ edges. We have devised a method to calculate $\Gamma_{\Graph{G}}(\Clusters,\Edges)$ from Monte Carlo simulations.