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{
    "pk": 65027,
    "title": "Some enumerative properties of parking functions",
    "subtitle": null,
    "abstract": "A parking function is a sequence \\((\\pi_1,\\dots, \\pi_n)\\) of positive integers such that if \\(\\lambda_1\\leq\\cdots\\leq \\lambda_n\\) is the increasing rearrangement of \\(\\pi_1,\\dots,\\pi_n\\), then \\(\\lambda_i\\leq i\\) for \\(1\\leq i\\leq n\\). In this paper we obtain some new results on the enumeration of parking functions. We will consider the joint distribution of several sets of statistics on parking functions. The distribution of most of these individual statistics is known, but the joint distributions are new. Parking functions of length \\(n\\) are in bijection with labelled forests on the vertex set \\([n]=\\{1,2,\\dots,n\\}\\) (or rooted trees on \\([n]_0=\\{0,1,\\dots,n\\}\\) with root \\(0\\)), so our results can also be applied to labelled forests. Extensions of our techniques are discussed, including an extension to a probabilistic scenario.\n \nMathematics Subject Classifications: 05A15, 60C05, 05A19\n \nKeywords: Parking function, labelled forest, generating function, recurrence, Pollak's circle argument",
    "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": "Parking function"
        },
        {
            "word": "labelled forest"
        },
        {
            "word": "generating function"
        },
        {
            "word": "recurrence"
        },
        {
            "word": "Pollak's circle argument"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/9862s2dg",
    "frozenauthors": [
        {
            "first_name": "Richard",
            "middle_name": "P.",
            "last_name": "Stanley",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of Miami, Coral Gables, FL 33124, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Mei",
            "middle_name": "",
            "last_name": "Yin",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of Denver, Denver, CO 80208, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2026-02-02T10:09:47Z",
    "date_accepted": "2026-02-02T10:09:47Z",
    "date_published": "2025-12-20T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65027/galley/49837/download/"
        }
    ]
}