Combinatorial Product

Let and be Combinatorial Structures.
We define their product as:

with weight

Lemma

Let and be Combinatorial Structures.
Then the Exponential Generating Function satisfies:

Proof

Let

Then the combined weight of objects of size in is

Thus

which matches the product in the Formal Power Series Ring so:

Lemma

Let and be Unlabelled Structures.
Then the Ordinary Generating Function satisfies:

Proof

The combined weight of objects of size in is:

and thus the result follows.