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

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).