Reflective numerical semigroups

TL;DR

This paper introduces the concept of reflective numerical semigroups, characterizes them explicitly, and provides formulas for counting such semigroups based on genus or Frobenius number.

Contribution

It defines reflective numerical semigroups, characterizes all of them explicitly, and derives formulas for their enumeration by genus and Frobenius number.

Findings

Explicit description of all reflective numerical semigroups

Formulas for counting reflective semigroups by genus

Identification of reflective members in known families

Abstract

We define a reflective numerical semigroup of genus $g$ as a numerical semigroup that has a certain reflective symmetry when viewed within $\mathbb{Z}$ as an array with $g$ columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo $g$. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.