{"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/counting-numerical-semigroups-by-frobenius","title":"Counting numerical semigroups by Frobenius number, multiplicity, and depth","arxiv_id":"2208.14587","date":"2022-08-31","proceeding":null,"authors":["Sean Li"],"abstract":"In 1990, Backelin showed that the number of numerical semigroups with Frobenius number $f$ approaches $C_i \\cdot 2^{f/2}$ for constants $C_0$ and $C_1$ depending on the parity of $f$. In this paper, we generalize this result to semigroups of arbitrary depth by showing there are $\\lfloor{(q+1)^2/4}\\rfloor^{f/(2q-2)+o(f)}$ semigroups with Frobenius number $f$ and depth $q$. More generally, for fixed $q \\geq 3$, we show that, given $(q-1)m < f < qm$, the number of numerical semigroups with Frobenius number $f$ and multiplicity $m$ is\\[\\left(\\left\\lfloor \\frac{(q+2)^2}{4} \\right\\rfloor^{\\alpha/2} \\left \\lfloor \\frac{(q+1)^2}{4} \\right\\rfloor^{(1-\\alpha)/2}\\right)^{m + o(m)}\\] where $\\alpha = f/m - (q-1)$. Among other things, these results imply Backelin's result, strengthen bounds on $C_i$, characterize the limiting distribution of multiplicity and genus with respect to Frobenius number, and resolve a recent conjecture of Singhal on the number of semigroups with fixed Frobenius number and maximal embedding dimension.","url_abs":"https://arxiv.org/abs/2208.14587v2","url_pdf":"https://arxiv.org/pdf/2208.14587v2.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":"counting-numerical-semigroups-by-frobenius","repo_url":"https://github.com/zhdag/stressed","is_official":0,"mentioned_in_paper":0,"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}