{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/sub-fibonacci-behavior-in-numerical-semigroup","title":"Sub-Fibonacci behavior in numerical semigroup enumeration","arxiv_id":"2202.05755","date":"2022-02-11","proceeding":null,"authors":["Daniel G. Zhu"],"abstract":"In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most $3$ and that the number $n_g$ of numerical semigroups of a genus $g$ is asymptotic to $S\\varphi^g$, where $S$ is some positive constant and $\\varphi \\approx 1.61803$ is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause $n_g$ to deviate from a perfect exponential, including the number of semigroups with depth at least $4$. Among other applications, these results imply the sharpest known asymptotic bounds on $n_g$ and shed light on a conjecture by Bras-Amor\\'os (2008) that $n_g \\geq n_{g-1} + n_{g-2}$. Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.","url_abs":"https://arxiv.org/abs/2202.05755v3","url_pdf":"https://arxiv.org/pdf/2202.05755v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"sub-fibonacci-behavior-in-numerical-semigroup","repo_url":"https://github.com/zhdag/stressed","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}