of the objects, then the OGF of the configurations is given by the substituted cycle index In the labelled case we use an exponential generating function (EGF) g(z) of the objects and apply the Labelled enumeration theorem, which says that the EGF of the configurations is given by We are able to enumerate filled slot configurations using either PET in the unlabelled case or the labelled enumeration theorem in the labelled case. We now ask about the generating function of configurations obtained when there is more than one set of slots, with a permutation group acting on each. Clearly the orbits do not intersect and we may add the respective generating functions. Suppose, for example, that we want to enumerate unlabelled sequences of length two or three of some objects contained in a set X. There are two sets of slots, the first one containing two slots, and the second one, three slots. The group acting on the first set is , and on the second slot, . We represent this by the following formal power series in X: where the term is used to denote the set of orbits under G and , which denotes in the obvious way the process of distributing the objects from X with repetition into the n slots. Similarly, consider the labelled problem of creating cycles of arbitrary length from a set of labelled objects X. This yields the following series of actions of cyclic groups: Clearly we can assign meaning to any such power series of quotients (orbits) with respect to permutation groups, where we restrict the groups of degree n to the conjugacy classes of the symmetric group , which form a unique factorization domain. (The orbits with respect to two groups from the same conjugacy class are isomorphic.) This motivates the following definition. A class of combinatorial structures is a formal series where (the "A" is for "atoms") is the set of primes of the UFD and In the following we will simplify our notation a bit and write e.g. for the classes mentioned above. The Flajolet–Sedgewick fundamental theorem A theorem in the Flajolet–Sedgewick theory of symbolic combinatorics treats the enumeration problem of labelled and unlabelled combinatorial classes by means of the creation of symbolic operators that make it possible to translate equations involving combinatorial structures directly (and automatically) into equations in the generating functions of these structures. Let be a