Computing the number of numerical semigroups using generating functions

TL;DR

This paper introduces a novel generating function approach to efficiently count numerical semigroups with specified genus or Frobenius number, providing theoretical complexity results and explicit formulas for certain cases.

Contribution

It develops a new methodology using generating functions to count numerical semigroups, including complexity analysis and explicit formulas for specific multiplicities.

Findings

Polynomial-time complexity results for counting semigroups

Explicit formulas for multiplicity 3 and 4 cases

Effective generating function methodology for enumeration

Abstract

This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting the number of these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.