Combinatorial classes, counting sequences and ordinary generating functions

A combinatorial class is a pair $(\mathcal{A},\mathrm{\mid }\cdot \mathrm{\mid })$ consisting of a set $\mathcal{A}$ and a size map $\mathrm{\mid }\cdot \mathrm{\mid }:\mathcal{A}\to \mathbb{N}$ such that, for every $n\in \mathbb{N}$, the level

${\mathcal{A}}_n:=\{a\in \mathcal{A}:\mathrm{\mid }a\mathrm{\mid }=n\}$

is finite (The cardinality $\mid A\mid$ of a finite set).

The counting sequence of $\mathcal{A}$ is

$a_n:=\mathrm{\mid }{\mathcal{A}}_n\mathrm{\mid }(n\in \mathbb{N}),$

Viewing each natural coefficient $a_n$ as its canonical integer by The naturals embed in the integers, its ordinary generating function is the formal power series

$A(x):={\sum }_{n\ge 0}{a}_n{x}^n\in \mathbb{Z}\mathrm{⟦}x\mathrm{⟧}$

in the sense of Formal power series over a commutative ring and the coefficient-extraction functional $[x^n]$.

An isomorphism of combinatorial classes $\mathcal{A}\to \mathcal{B}$ is a bijection $f:\mathcal{A}\to \mathcal{B}$ such that $\mathrm{\mid }f(a)\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }$ for every $a\in \mathcal{A}$. Such a bijection identifies each level ${\mathcal{A}}_n$ with the corresponding level ${\mathcal{B}}_n$, so isomorphic classes have the same counting sequence and the same ordinary generating function.