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{
    "pk": 64827,
    "title": "Toppleable permutations, excedances and acyclic orientations",
    "subtitle": null,
    "abstract": "Recall that an excedance of a permutation $\\pi$ is any position $i$ such that $\\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\\{1,\\dots, \\lfloor (n-1)/2 \\rfloor\\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph $K_{\\lceil n/2 \\rceil, \\lfloor n/2 \\rfloor + 1}$. We also give a formula for the number of AUSOs of complete multipartite graphs. We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.\n \nMathematics Subject Classifications: 05A19, 05A05, 05C30",
    "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": "Toppleable permutations"
        },
        {
            "word": "acyclic orientations"
        },
        {
            "word": "excedances"
        },
        {
            "word": "collapsed permutations"
        },
        {
            "word": "complete bipartite"
        },
        {
            "word": "complete multipartite"
        },
        {
            "word": "Genocchi numbers"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/08z5b229",
    "frozenauthors": [
        {
            "first_name": "Arvind",
            "middle_name": "",
            "last_name": "Ayyer",
            "name_suffix": "",
            "institution": "Department of Mathematics, Indian Institute of Science, Bangalore 560012, India.",
            "department": ""
        },
        {
            "first_name": "Daniel",
            "middle_name": "",
            "last_name": "Hathcock",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Prasad",
            "middle_name": "",
            "last_name": "Tetali",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2022-03-23T20:21:06Z",
    "date_accepted": "2022-03-23T20:21:06Z",
    "date_published": "2022-03-31T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64827/galley/49637/download/"
        }
    ]
}