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

In a text 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 is $X$ and $Y$ are finite sets:

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

If $|X|=|Y|$, then $|\mathcal{A}_X|=|\mathcal{A}_Y|$.

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

Luis Alexandher
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  • Crosspost at https://mathoverflow.net/questions/413622/confusion-in-definition-of-class-of-structures-and-combinatorial-class – Sean Eberhard Jan 12 '22 at 12:17

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They seem different. The first definition seems to assume we have canonical represtations of the things we're counting (e.g. for the class of permutations we might have $\mathcal A_n=S_n$), whereas the second definition allows for multiple representations (so e.g. $\mathcal A_X$ might denote the symmetric group on $X$). I don't think these technicalities are too important when you're actually doing combinatorics as long as you understand what you're trying to count.

Karl
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  • The problem is that I need to identify each definition well since I need them for my thesis and for now I can't find a good explanation. – Luis Alexandher Jan 03 '22 at 22:52
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    I'd suggest looking carefully at how both definitions are used to model a specific counting problem in order to get comfortable with how each one works. There's nothing deep about the definitions themselves. – Karl Jan 05 '22 at 17:08