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{
    "pk": 64913,
    "title": "Counting numerical semigroups by Frobenius number, multiplicity, and depth",
    "subtitle": null,
    "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.\n \nMathematics Subject Classifications: 05A16, 20M14\n \nKeywords: Numerical semigroups, Kunz coordinates, graph homomorphisms",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Numerical semigroups"
        },
        {
            "word": "Kunz coordinates"
        },
        {
            "word": "graph homomorphisms"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/38w3q7f8",
    "frozenauthors": [
        {
            "first_name": "Sean",
            "middle_name": "",
            "last_name": "Li",
            "name_suffix": "",
            "institution": "Department of Mathematics, Massachusetts Institute of Technology, Cambridge, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2023-12-22T13:40:03Z",
    "date_accepted": "2023-12-22T13:40:03Z",
    "date_published": "2023-12-22T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64913/galley/49723/download/"
        }
    ]
}