Algorithms for numerical semigroups with fixed maximum primitive

TL;DR

This paper introduces an algorithm to analyze numerical semigroups with a fixed maximum primitive, enabling counting and verification of Wilf's conjecture for semigroups with maximum primitive up to 60.

Contribution

The paper presents a novel algorithm for exploring properties of numerical semigroups with a fixed maximum primitive, including enumeration and conjecture verification.

Findings

No counterexamples to Wilf's conjecture found up to maximum primitive 60.

Efficient enumeration of numerical semigroups with a given maximum primitive.

Verification of Wilf's conjecture for a new class of semigroups.

Abstract

We present an algorithm to explore various properties of the numerical semigroups with a given maximum primitive. In particular, we count the number of such numerical semigroups and verify that there is no counterexample to Wilf's conjecture among the numerical semigroups with maximum primitive up to \(60\).