On the integer partitions recursive structure

TL;DR

This paper explores the recursive structure of integer partitions, revealing how they can be decomposed into polynomial and quasiperiodic components, and how weights relate to smaller partitions.

Contribution

It introduces a novel perspective on integer partitions by analyzing their recursive structure through Sylvester waves and polynomial components.

Findings

Partition weights are sums of smaller partitions.

Integer partitions can be decomposed into polynomial and quasiperiodic parts.

Recursive structure links partitions to smaller sets.

Abstract

Sylvester showed that the partition of an integer into a set of positive integers can be represented as a sum of the polynomial term and quasiperiodic components called the Sylvester waves. The wave itself is a weighted sum of the polynomial terms multiplied by the periodic functions. The integer weights are found to be a sum of partitions into a smaller set of integers implying the recursive structure of integer partitions.