Counting Numerical Semigroups

Abstract

A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of a conjecture of Bras-Amorós that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.

arXiv comment14 pages. To appear in American Mathematical Monthly special issue on research by undergraduates

References in corpus (9)

• numericalsgps, a GAP package for numerical semigroups
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• Acute Semigroups, the Order Bound on the Minimum Distance and the Feng-Rao Improvements
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Cited by in corpus (6)

• Gapsets and numerical semigroups
• Numerical Sets, Core Partitions, and Integer Points in Polytopes
• Numerical semigroups, polyhedra, and posets I: the group cone
• On the genus of a quotient of a numerical semigroup
• A generalization of a theorem about gapsets with depth at most three
• Counting numerical semigroups by genus and even gaps via Kunz-coordinate vectors