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Let $A_n$ be a family of sequences $\{a_i\}_{i=1}^n$ of length $n$. I'll refer to sequence elements of $A_n$ as $a$. Then define

$$G(z):=\sum_{a\in A_n}\prod_{i=1}^n(z+a_i).$$

Here's one possible concrete example. Let $A_n=S_n$, the permutation group on $n$ elements. This is a rather easy example because the product term is the same for all elements of $S_n$ and we get the nice expression $G(z)=n!z^{[n]}$ where $z^{[n]}:=(z+1)\cdots (z+n)$.

I'm interested in what information can be extracted from $G(z)$ about $A_n$. In particular, is there a common name or reference for the family of such generating functions of this sum-product form?

For example:


In other words we can extract the size of $A_n$.

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