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{
    "pk": 65020,
    "title": "Skeletal generalizations of chip-firing games, parking functions, and Dyck paths",
    "subtitle": null,
    "abstract": "For \\(0\\leq k\\leq n-1\\), we introduce a family of \\(k\\)-skeletal paths which are counted by the \\(n\\)th Catalan number for each \\(k\\), and specialize to Dyck paths when \\(k=n-1\\). We similarly introduce \\(k\\)-skeletal parking functions which are equinumerous with spanning trees on the complete graph with \\(n+1\\) vertices for each \\(k\\), and specialize to classical parking functions for \\(k=n-1\\). The preceding constructions are generalized to paths lying in a trapezoid with base \\(c › 0\\) and southeastern diagonal of slope \\(1/m\\); \\(c\\) and \\(m\\) need not be integers. We give bijections among these families when \\(k\\) varies with \\(m\\) and \\(c\\) fixed. Our constructions are motivated by chip firing and have connections to combinatorial representation theory and tropical geometry.\n \nMathematics Subject Classifications: 05A15, 05A19, 05C57\n \nKeywords: Chip firing, skeletal objects, lattice paths, Dyck paths,   parking functions, Catalan numbers, ballot numbers",
    "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": "Chip firing"
        },
        {
            "word": "skeletal objects"
        },
        {
            "word": "lattice paths"
        },
        {
            "word": "Dyck paths"
        },
        {
            "word": "parking functions"
        },
        {
            "word": "Catalan numbers"
        },
        {
            "word": "ballot numbers"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/99g5z77h",
    "frozenauthors": [
        {
            "first_name": "Spencer",
            "middle_name": "",
            "last_name": "Backman",
            "name_suffix": "",
            "institution": "Dept. of Mathematics and Statistics, University of Vermont, Burlington, Vermont, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Cole",
            "middle_name": "",
            "last_name": "Charbonneau",
            "name_suffix": "",
            "institution": "Unaffiliated",
            "department": ""
        },
        {
            "first_name": "Nicholas",
            "middle_name": "A.",
            "last_name": "Loehr",
            "name_suffix": "",
            "institution": "Department of Mathematics, Virginia Tech, Blacksburg, Virginia, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Patrick",
            "middle_name": "",
            "last_name": "Mullins",
            "name_suffix": "",
            "institution": "Unaffiliated",
            "department": ""
        },
        {
            "first_name": "Mazie",
            "middle_name": "",
            "last_name": "O'Connor",
            "name_suffix": "",
            "institution": "Dept. of Mathematics and Statistics, University of Vermont, Burlington, Vermont, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Gregory",
            "middle_name": "S.",
            "last_name": "Warrington",
            "name_suffix": "",
            "institution": "Dept. of Mathematics and Statistics, University of Vermont, Burlington, Vermont, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2026-02-02T09:14:33Z",
    "date_accepted": "2026-02-02T09:14:33Z",
    "date_published": "2025-12-20T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65020/galley/49830/download/"
        }
    ]
}