Simple proofs and expressions for the restricted partition function and its polynomial part

TL;DR

This paper presents a straightforward algebraic approach to deriving formulas for the restricted partition function and its polynomial component, simplifying previous methods and providing clearer insights.

Contribution

It introduces a simple derivation method using elementary algebra and a lemma to express the polynomial part as an average, enhancing understanding of partition functions.

Findings

Derived explicit formulas for the restricted partition function.

Provided a new elementary algebraic proof approach.

Expressed the polynomial part as an average of the partition function.

Abstract

In this note, we provide a simple derivation of expressions for the restricted partition function and its polynomial part. Our proof relies on elementary algebra on rational functions and a lemma that expresses the polynomial part as an average of the partition function.