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: