On compositions of natural numbers

TL;DR

This paper introduces compositions of natural numbers, explores restricted forms such as odd parts, derives counting formulas for these forms, and discusses their properties in an expository manner.

Contribution

It provides formulas for counting restricted compositions of natural numbers, focusing on specific forms like odd parts, and offers an accessible overview of the topic.

Findings

Derived formulas for counting compositions with restrictions

Analyzed compositions with odd parts and other simple restrictions

Provided an expository discussion 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.