Unrefinable Partitions into Distinct Parts and Numerical Semigroups

TL;DR

This paper explores the relationship between unrefinable partitions into distinct parts and numerical semigroups, establishing new criteria and classifications through analysis of Young diagrams and hooksets.

Contribution

It introduces a novel connection between unrefinable partitions and numerical semigroups, extending classifications and characterising partitions with maximal missing parts.

Findings

Certain unrefinable partitions correspond to symmetric numerical semigroups when the maximal part is prime.

New criteria for recognising unrefinable partitions based on hooksets are established.

Structural decompositions of unrefinable partitions are provided, with implications for maximal unrefinable partitions.

Abstract

This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising unrefinable partitions are established. A correspondence between missing parts and the gaps of numerical semigroups is developed, extending previous classifications and enabling the characterisation of partitions with maximal numbers of missing parts. In particular, the results show that certain families of unrefinable partitions correspond precisely to symmetric numerical semigroups when the maximal part is prime. Further structural consequences, examples, and a decomposition of unrefinable partitions by minimal excludant are discussed, together with implications for the study of maximal unrefinable partitions.