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{
    "pk": 64890,
    "title": "An asymptotically tight lower bound for superpatterns with small alphabets",
    "subtitle": null,
    "abstract": "A permutation \\(\\sigma \\in S_n\\) is a \\(k\\)-superpattern (or \\(k\\)-universal) if it contains each \\({\\tau \\in S_k}\\) as a pattern. This notion of \"superpatterns\" can be generalized to words on smaller alphabets, and several questions about superpatterns on small alphabets have recently been raised in the survey of Engen and Vatter. One of these questions concerned the length of the shortest \\(k\\)-superpattern on \\([k+1]\\). A construction by Miller gave an upper bound of \\({(k^2+k)/2}\\), which we show is optimal up to lower-order terms. This implies a weaker version of a conjecture by Eriksson, Eriksson, Linusson and Wastlund. Our results also refute a 40-year-old conjecture of Gupta.\n \nMathematics Subject Classifications: 05A05, 60C05\n \nKeywords: Patterns, permutations, probabalistic method",
    "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": "Patterns"
        },
        {
            "word": "permutations"
        },
        {
            "word": "probabalistic method"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/0cd5r5c8",
    "frozenauthors": [
        {
            "first_name": "Zach",
            "middle_name": "",
            "last_name": "Hunter",
            "name_suffix": "",
            "institution": "ETH Zurich, Switzerland",
            "department": ""
        }
    ],
    "date_submitted": "2023-09-13T15:12:02Z",
    "date_accepted": "2023-09-13T15:12:02Z",
    "date_published": "2023-09-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64890/galley/49700/download/"
        }
    ]
}