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{
    "pk": 64938,
    "title": "A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's \\(G_2\\) spider",
    "subtitle": null,
    "abstract": "We prove a conjectured asymptotic formula of Kuperberg from the representation theory of the Lie algebra \\(G_2\\). Given two non-negative integer sequences \\((a_n)_{n\\geq 0}\\) and \\((b_n)_{n\\geq 0}\\), with \\(a_0=b_0=1\\), it is well-known that if the identity \\(B(x)=A(xB(x))\\) holds for the generating functions \\(A(x)=1+\\sum_{n\\geq 1} a_n x^n\\) and \\(B(x)=1+\\sum_{n\\geq 1} b_n x^n\\), then \\(b_n\\) is the number of rooted planar trees with \\(n+1\\) vertices such that each vertex having \\(i\\) children may be colored with any one of \\(a_i\\) distinct colors. Kuperberg proved a specific case when this identity holds, namely when \\(b_n=\\dim \\operatorname{Inv}_{G_2} (V(\\lambda_1)^{\\otimes n})\\), where \\(V(\\lambda_1)\\) is the 7-dimensional fundamental representation of \\(G_2\\), and \\(a_n\\) is the number of triangulations of a regular \\(n\\)-gon such that each internal vertex has degree at least \\(6\\). He also observed that \\(\\limsup_{n\\to\\infty}\\sqrt[n]{a_n}\\leq 7/B(1/7)\\) and conjectured that this estimate is sharp, or, in terms of power series, that the radius of convergence of \\(A(x)\\) is exactly \\(B(1/7)/7\\). We prove this conjecture by introducing a new criterion for sharpness in the analogous estimate for general power series \\(A(x)\\) and \\(B(x)\\) satisfying \\(B(x)=A(xB(x))\\). Moreover, by way of singularity analysis performed on a recently discovered generating function for \\(B(x)\\), we significantly refine the conjecture by deriving an asymptotic formula for the sequence \\((a_n)\\).\n \nMathematics Subject Classifications: 05A16, 05E10\n \nKeywords: Analytic combinatorics",
    "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": "Analytic combinatorics"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3t4812dw",
    "frozenauthors": [
        {
            "first_name": "Robert",
            "middle_name": "",
            "last_name": "Scherer",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of California, Davis, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2024-07-01T16:25:08+02:00",
    "date_accepted": "2024-07-01T16:25:08+02:00",
    "date_published": "2024-06-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
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            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64938/galley/49748/download/"
        }
    ]
}