New Eliahou Semigroups and Verification of the Wilf Conjecture for Genus up to 65

TL;DR

This paper introduces a graphical reinterpretation of the seeds algorithm to explore numerical semigroups, uses it to identify all Eliahou semigroups up to genus 65, and confirms the Wilf conjecture for these cases.

Contribution

It presents a novel graphical approach to the seeds algorithm and verifies the Wilf conjecture for all Eliahou semigroups up to genus 65.

Findings

All Eliahou semigroups up to genus 65 satisfy the Wilf conjecture.

The seeds algorithm was effectively used to explore the tree of numerical semigroups.

The graphical reinterpretation enhances understanding of the seeds algorithm.

Abstract

We give a graphical reinterpretation of the seeds algorithm to explore the tree of numerical semigroups. We then exploit the seeds algorithm to find all the Eliahou semigroups of genus up to 65. Since all these semigroups satisfy the Wilf conjecture, this shows that the Wilf conjecture holds up to genus 65.