I understand that a combinatorial class $\mathcal{A}$ is a set of objects, with a function of size $\lvert\cdot\rvert\_{\mathcal{A}}:\mathcal{A}\to \mathbb{N}$. With objects of size $n$: $\mathcal{A}\_n=\{\alpha\in\mathcal{A}\;:\; \lvert\alpha\rvert\_{\mathcal{A}}=n\}$, and a condition of finitude $\#\mathcal{A}\_n<\infty$.


In a book ["Introduction to Combinatorial Enumeration" by David G. Wagner](https://uwaterloo.ca/combinatorics-and-optimization/sites/ca.combinatorics-and-optimization/files/uploads/files/co330-notes.pdf) chapter 11, I came across the term "class of structures", and they define it as:


A class $\mathcal{A}$ associated with a finite set $X$ is another finite set $\mathcal{A}\_X$, such that if $X$ and $Y$ are finite sets:


If $X\neq Y$ then $\mathcal{A}\_X\cap \mathcal{A}\_Y=\emptyset$.


If $X$ and $Y$ are finite sets with $\lvert X\rvert=\lvert Y\rvert$, then $\lvert\mathcal{A}\_X\rvert=\lvert\mathcal{A}\_Y\rvert$.


My question is whether these two definitions refer to the same thing. I am somewhat confused.



Confusion in definition of class of structures and combinatorial class
Part of the problem is that you have stated the definition of "class of structures" incorrectly.
(Wagner states it correctly.) The first sentence should be: "A class $\mathcal{A}$ of structures associates to every finite set $X$ another finite set $\mathcal{A}\_X$, in such a way that the following two conditions are satisfied:"


The two definitions represent different concepts. The first refers to "unlabeled objects" like partitions, binary words, or lattice paths, where the set of objects under consideration of size $n$ is finite. The second refers to "labeled objects" like graphs or permutations. For example, if $\mathcal{A}$ represents the class of graphs, then for any finite set $X$, $\mathcal{A}\_X$ is the set of graphs with vertex set $X$. Roughly speaking, "combinatorial classes" correspond to ordinary generating functions and "classes of structures" correspond to exponential generating functions.


