Counting Numerical Semigroups
Combinatorics
2022-01-25 v1
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 ? We outline Zhai's proof of a conjecture of Bras-Amor\'os 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.
Cite
@article{arxiv.1707.02551,
title = {Counting Numerical Semigroups},
author = {Nathan Kaplan},
journal= {arXiv preprint arXiv:1707.02551},
year = {2022}
}
Comments
14 pages. To appear in American Mathematical Monthly special issue on research by undergraduates