Simple alternative to the Hardy-Ramanujan-Rademacher formula for p(N)

TL;DR

This paper introduces a simple, closed-form alternative to the Hardy-Ramanujan-Rademacher formula for calculating the partition function p(N), based on analyzing the structure of the integer partitions tree.

Contribution

It provides a novel combinatorial approach that counts paths in the partitions tree to derive a transparent formula for p(N), bypassing complex existing formulas.

Findings

Derived a closed-form formula for the number of partitions into M parts

Summed over M to obtain an alternative to the Hardy-Ramanujan-Rademacher formula

Validated the formula's effectiveness through combinatorial analysis

Abstract

A recent paper examined the global structure of integer partitions sequences and, via combinatorial analysis using modular arithmetic, derived a closed form expression for a map from (N, M) to the set of all partitions of a positive integer N into exactly M positive integer summands. The output of the IPS map was a "matrix" having M columns and a number of rows equal to p[N, M], the number of partitions of N into M parts. The global structure of integer partition sequences (IPS) is that of a complex tree. In this paper, we examine the structure of the IPS tree and, by counting the number of directed paths through the tree, obtain a simple formula which gives, in closed form, the total number of partitions of N into exactly M parts. By summing over M, we obtain a transparent alternative to the Hardy-Ramanujan-Rademacher formula for p(N).