Stochastic estimations of a total number of classes for the clusterings with too enormous samples to be accommodate into a clustering engine

We considered the problem how to handle the exploding number of possibilities to be sorted into irreducible classes by using a clustering tool when its input capacity cannot accommodate the total number of the possibility. Concrete situations are explained taking examples of atomic substitutions in the supercell modeling of alloys. The number of the possibility sometimes amounts to $\sim$ trillion being too large to be accommodate. It is hence not practically feasible to identify how many irreducible classes exist by straightforward manner even though there are several tools available to perform the clustering. We have developed a stochastic framework to avoid the shortage of capacity, providing a method to estimate the total number of irreducible classes (the order of the classes) as a statistical estimate. A prominent conclusion derived here is that the statistical variation of the number of classes at each sampling trial is working as a promising measure to estimate the order.