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{
    "pk": 64818,
    "title": "On the Okounkov-Olshanski formula for standard tableaux of skew shapes",
    "subtitle": null,
    "abstract": "The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two $q$-analogues for reverse plane partitions, which complement work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.\n \nMathematics Subject Classifications: 05A15, 05A15, 05A19, 05A05",
    "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": "Standard tableaux"
        },
        {
            "word": "semistandard tableaux"
        },
        {
            "word": "skew shapes"
        },
        {
            "word": "lozenge tilings"
        },
        {
            "word": "reverse plane partitions"
        },
        {
            "word": "Knutson-Tao puzzles"
        },
        {
            "word": "flagged tableaux"
        },
        {
            "word": "excited diagrams"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/9zx5x4ck",
    "frozenauthors": [
        {
            "first_name": "Alejandro",
            "middle_name": "H.",
            "last_name": "H. Morales",
            "name_suffix": "",
            "institution": "Department of Mathematics and Statistics, UMass, Amherst, MA 01003, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Daniel",
            "middle_name": "G.",
            "last_name": "G. Zhu",
            "name_suffix": "",
            "institution": "Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2022-03-23T18:53:25Z",
    "date_accepted": "2022-03-23T18:53:25Z",
    "date_published": "2022-03-31T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64818/galley/49628/download/"
        }
    ]
}