Definition 10.1. [BLL98, Def. 1.1.4] Consider two F-structures s1 ∈ F[U] and s2 ∈ F[V ]. A bijection σ : U → V is called an isomorphism of s1 to s2 if s2 = σ⋅s1 = F[σ](s1). One says that these structures have the same isomorphism type. Moreover, an isomorphism from s to s is said to be an automorphism of s.
Definition 10.2. [BLL98, p. 14] Let U be a set. We define an equivalence relation ∼ on F[U] by setting for s,t ∈ F[U]
s ∼ t if and only if s and t have the same isomorphism type.
In other words, s ∼ t if and only if there exists a permutation π : U → U such that
F[π](s) = t. By 
we denote the isomorphism type of s.
Let us furthermore define
| T(Fn) | = F[n] ∕ ∼ | (120) |
| T(F) | = ∑ n≥0T(Fn). | (121) |
Note that 
= Sn(t) is also know as the orbit of t with respect to the action
![α : Sn × F [n] → F[n], (σ,s) ↦→ F [σ ](s)](combinat225x.png) | (122) |
Definition 10.3. [BLL98, p. 15] The (isomorphism) type generating series or ordinary generating series of a species of structures F is the formal power series
 | (123) |
= card(T(Fn)) is the number of (unlabelled) F-structures on a set U
of n elements.
Usage
f: OrdinaryGeneratingSeries := monom;
Description
Ordinary generating series.
OrdinaryGeneratingSeries is the domain that represents ordinary generating series, i. e., formal power series f of the form
 | (124) |
Exports of OrdinaryGeneratingSeries
FormalPowerSeriesCategory Integer;
count: (%, MachineInteger) -> Integer Counts the number of structures of a given size.
Export of OrdinaryGeneratingSeries
count: (%, MachineInteger) -> Integer
Description
Counts the number of structures of a given size.