On compositions of natural numbers

Abstract

In this expository note, we introduce the reader to compositions of a natural number, e.g., $2+1+2+1+7+1$ is a composition of 14, and $1+2$ and $2+1$ are two different compositions of 3. We discuss some simple restricted forms of compositions, e.g., $23+17+33$ is a composition of 73 into three odd parts. We derive formulas that count the number of so restricted forms of compositions of a natural number $n$, and we conclude with a brief general discussion of the topic.