On the smallest partition associated to a numerical semigroup

TL;DR

This paper investigates the minimal size of partitions whose hook lengths complement a given numerical semigroup, addressing a specific combinatorial problem related to hook length sets.

Contribution

It introduces the problem of determining the smallest partition with a hook length set complement of a numerical semigroup, a novel focus in partition theory.

Findings

Formulated the problem of finding the smallest partition with a given hook length complement.

Connected hook length sets of partitions to numerical semigroups.

Provided initial insights or results on the minimal partition size for certain semigroups.

Abstract

The set of hook lengths of an integer partition $\lambda$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about the distribution of sizes of this finite set of partitions. We focus on the problem of determining the size of the smallest partition with its set of hook lengths equal to $\mathbb{N}\setminus S$.