Unlabelled Composition

Let and be Unlabelled Structures.
Their unlabelled composition is defined as:

where and

as the Combinatorial Product.

This is only well defined if ,
otherwise there is an infinite contribution from partitions containing empty sets.

Lemma

Let and have Ordinary Generating Functions and .
Then

Proof

Firstly

as the Combinatorial Product.
Now suppose

The combined weight of objects of size in is

We conclude:

^ cheeky infinite sum swap but its formal infinite sum.