Verification and generation of unrefinable partitions

TL;DR

This paper explores the properties of unrefinable partitions, providing algorithms for testing and enumerating such partitions, and analyzes their computational complexity.

Contribution

It introduces algorithms for verifying and enumerating unrefinable partitions, advancing understanding of their algorithmic properties.

Findings

Developed algorithms for testing unrefinability

Created methods for enumerating unrefinable partitions

Analyzed the complexity of these algorithms

Abstract

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, being an unrefinable partition means that none of the parts can be written as the sum of smaller integers without introducing a repetition. We address the algorithmic aspects of unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given integer. We design two algorithms to solve the two mentioned problems and we discuss their complexity.