Constructing Numerical Semigroups of a Given Genus

TL;DR

This paper introduces a new, simplified method for constructing and counting numerical semigroups of a given genus, leading to improved asymptotic bounds and bypassing traditional semigroup tree analysis.

Contribution

It presents a novel approach for directly constructing numerical semigroups of a fixed genus, improving bounds and avoiding complex existing methods.

Findings

Provides a new construction method for numerical semigroups

Offers an improved asymptotic lower bound for n_g

Circumvents analysis of the semigroup tree

Abstract

Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros conjectured that n_g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for n_g.