Finite Sequence

Combinatorial Structure of finite sequences is defined by

where is the Combinatorial Structure of singletons:

When is a Combinatorial Structure,
we may write for Combinatorial Composition.
When is an Unlabelled Structure,
we may write for Unlabelled Composition.

Lemma

Ordinary Generating Function and Exponential Generating Function are:

Proof 1

Both go well with Combinatorial Union and Combinatorial Product
and we have

Proof 2

Each sequence is either empty or has a first element thus: