Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)

TL;DR

This paper introduces a Maple package called PARTITIONS that automatically derives and proves explicit formulas for the number of partitions of n into at most m parts, showcasing the effectiveness of the quasi-polynomial ansatz.

Contribution

The paper presents a novel Maple tool that automates the discovery and proof of partition formulas using the quasi-polynomial approach.

Findings

Successfully automates derivation of partition formulas

Demonstrates power of rigorous guessing with quasi-polynomials

Provides explicit formulas for various m

Abstract

The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of "rigorous guessing" as facilitated by the quasi-polynomial ansatz.