Confusion in definition of class of structures and combinatorial class [closed]

Question

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 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.

Answer 1

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.