
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 as a numerical semigroup that has a certain reflective symmetry when viewed within as an array with columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo . 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.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Polynomial and algebraic computation
