Composition of an integer

Definition

Let $n$ be a natural number. A composition of $n$ is a finite sequence $(a_1, \dots, a_k)$ of positive integers with $a_1 + \dots + a_k = n$. The number $k$ is the number of parts of the composition.

Remarks

Unlike a partition, a composition records the order of its parts. The compositions of $3$ are $(3)$, $(2, 1)$, $(1, 2)$ and $(1, 1, 1)$. The number $0$ has exactly one composition, the empty sequence.

Explanations

L1 human

A composition of $n$ writes $n$ as an ordered sum of positive integers. So $2 + 1$ and $1 + 2$ are two different compositions of $3$.