Enumerating numerical sets associated to a numerical semigroup

TL;DR

This paper studies the enumeration of numerical sets associated with a numerical semigroup by introducing the void poset and characterizing the anti-atom problem through order ideals, especially for semigroups with small type.

Contribution

It introduces the void poset of a numerical semigroup and provides a bijection between numerical sets with a given atom monoid and order ideals, solving the anti-atom problem for small type semigroups.

Findings

Established a bijection between numerical sets and order ideals of the void poset.

Solved the anti-atom problem for numerical semigroups with small type.

Connected the problem to enumeration of integer partitions with specific hook lengths.

Abstract

A numerical set $T$ is a subset of $\mathbb N_0$ that contains $0$ and has finite complement. The atom monoid of $T$ is the set of $x \in \mathbb N_0$ such that $x+T \subseteq T$. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup $S$ and show that numerical sets with atom monoid $S$ are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when $S$ has small type.