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{
    "pk": 64978,
    "title": "Differential equations for the series of hypermaps with control on their full degree profile",
    "subtitle": null,
    "abstract": "We consider the generating series of oriented and non-oriented hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that these series have natural expansions in terms of Schur and Zonal symmetric functions, and with some particular specializations, they satisfy the celebrated KP and BKP equations. We prove that the full generating series of hypermaps satisfy a family of differential equations. We give a first proof which works for an \\(\\alpha\\) deformation of these series related to Jack polynomials. This proof is based on a recent construction formula for Jack characters using differential operators. We also provide a combinatorial proof for the orientable case. Our approach also applies to the series of \\(k\\)-constellations with control of the degrees of vertices of all colors. In other words, we obtain an equation for the generating function of Hurwitz numbers (and their \\(\\alpha\\)-deformations) with control of full ramification profiles above an arbitrary number of points. Such equations are new even in the orientable case.\n \nMathematics Subject Classifications: 05E05\n \nKeywords: Hypermaps, differential equations, Jack characters",
    "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": "Hypermaps"
        },
        {
            "word": "differential equations"
        },
        {
            "word": "Jack characters"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/6362t7q9",
    "frozenauthors": [
        {
            "first_name": "Houcine",
            "middle_name": "",
            "last_name": "Ben Dali",
            "name_suffix": "",
            "institution": "Université de Lorraine, CNRS, IECL, F-54000 Nancy, Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2025-03-14T20:13:08+03:00",
    "date_accepted": "2025-03-14T20:13:08+03:00",
    "date_published": "2025-03-15T10:00:00+03:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64978/galley/49788/download/"
        }
    ]
}