Fundamental theorem of combinatorial enumeration
The Fundamental Theorem of Combinatorial Enumeration is a theorem in combinatorics that solves the enumeration problem of labelled and unlabelled combinatorial classes. The unlabelled case is based on the Pólya enumeration theorem.
Classes of combinatorial structures
Consider the problem of distributing objects given by some generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration is the sum of the weights of the objects in the slots.
ThePólya enumeration theorem solves this problem in the unlabelled case. Let f(z) be the ordinary generating function (OGF) of the objects, then the OGF of the configurations is given by
