On integer partitions corresponding to numerical semigroups

TL;DR

This paper investigates the relationship between numerical semigroups and integer partitions, focusing on classifying and counting partitions that correspond to numerical semigroups based on various invariants.

Contribution

It characterizes and counts partitions arising from numerical semigroups using invariants like genus, Frobenius number, and multiplicity, extending previous work on the injection.

Findings

Count of partitions by genus, Frobenius number, and multiplicity.

Classification of partitions in the image of the injection.

Insights into the structure of partitions related to numerical semigroups.

Abstract

Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions.