The Fractal and The Recurrence Equations Concerning The Integer Partitions

TL;DR

The paper addresses counting integer partitions and deriving recurrence relations by introducing a fractal-style generator built on the ${\Omega(n)}$ decomposition and a parcel-cell framework. It defines a core result, ${\lambda = 2\tau - n}$, which governs how many child-parcels a parcel produces, enabling bijective counts of partition representations and leading to the pentagonal-number recurrence ${p(n)=p(n-1)+p(n-2)+p(n-12)-p(n-5)-p(n-7)}$ as a key example. By varying tails and applying the generator, it derives additional long-range recurrences involving past values such as ${p(n-6)}$, ${p(n-8)}$, ${p(n-20)}$, etc., illustrating an infinite family of identities and recovering known results like the pentagonal-number theorem as a special case. The work suggests a broader, self-similar framework for partition identities, with signed-sum expressions of ${n}$ in terms of ${p(i)}$ offering new structural insights and potential generalizations.

Abstract

This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the pentagonal number theorem.