Families of Eliahou semigroups linked to Farey intervals

TL;DR

This paper introduces new families of Eliahou semigroups linked to Farey intervals, explores their properties, and provides methods for their explicit construction, supporting the conjecture that they satisfy Wilf's conjecture.

Contribution

It defines and analyzes new families of Eliahou semigroups connected to Farey intervals, expanding previous classifications and offering construction techniques.

Findings

Most of these semigroups probably satisfy Wilf's conjecture.

A new representation and pruning method for the numerical semigroup tree was developed.

Conjecturally, all Eliahou semigroups of conductor up to 320 are included.

Abstract

We describe new families of Eliahou semigroups, encompassing previous families described by Delgado, Eliahou and Fromentin, and Bras-Amor\'os. A crucial parameter is a Farey interval associated to the semigroup. We show that these semigroups probably all satisfy Wilf's conjecture and describe ways to explicitly construct semigroups belonging to these families. This work is based on an exploration of the numerical semigroup tree giving (conjecturally) all Eliahou semigroups of conductor up to 320 thanks to a new way of representing the semigroups and pruning of unwanted branches.