Nuclear partitions and a formula for $p(n)$

TL;DR

This paper introduces a new approach to calculating the partition function p(n) using nuclear partitions, which are partitions with no parts equal to one, leading to novel formulas and congruences.

Contribution

It provides a simple formula for p(n) based on nuclear partitions and explores related variations, congruences, and parity applications.

Findings

Derived a formula for p(n) using nuclear partitions

Established Ramanujan-like congruences for p(n)

Applied the formula to analyze the parity of p(n)

Abstract

Define a "nuclear partition" to be an integer partition with no part equal to one. In this study we prove a simple formula to compute the partition function $p(n)$ by counting only the nuclear partitions of $n$, a vanishingly small subset by comparison with all partitions of $n$ as $n\to \infty$. Variations on the proof yield other formulas for $p(n)$, as well as Ramanujan-like congruences and an application to parity of the partition function.