On counting numerical semigroups by maximum primitive and Wilf's conjecture

TL;DR

This paper introduces a novel counting method for numerical semigroups based on their maximum primitive, explores its relation to Frobenius number counting, and shows most large semigroups satisfy Wilf's conjecture.

Contribution

It establishes a new counting approach for numerical semigroups, relates it to existing methods via M"obius transforms, and proves that almost all large semigroups satisfy Wilf's conjecture.

Findings

Counting by maximum primitive and Frobenius number are M"obius transforms of each other.

Almost all numerical semigroups with large maximum primitive satisfy Wilf's conjecture.

Semigroups with certain intersection size with (m, 2m) satisfy Wilf's conjecture.

Abstract

We introduce a new way of counting numerical semigroups, namely by their maximum primitive, and show its relation with the counting of numerical semigroups by their Frobenius number. We show that these two ways of counting are M\"obius transforms of one another. We also establish that almost all numerical semigroups with large enough maximum primitive satisfy Wilf's conjecture. A crucial step in the proof is a result of independent interest: a numerical semigroup $S$ with multiplicity $\mathrm{m}$ such that $|S\cap (\mathrm{m},2 \mathrm{m})|\geq \sqrt{2\mathrm{m}}$ satisfies Wilf's conjecture.