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{
    "pk": 64974,
    "title": "Combinatorics of rectangulations: old and new bijections",
    "subtitle": null,
    "abstract": "A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallels and differences between the weak and the strong cases. Then, we consider guillotine rectangulations, and prove that they can be characterized -- under all known mappings between permutations and rectangulations -- by avoidance of two mesh patterns that correspond to windmills in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.\n \nMathematics Subject Classifications: 05A05, 05A15, 05A16, 05C10, 06A07, 06B10\n \nKeywords: Rectangulations, combinatorial enumeration, generating functions, asymptotic enumeration, posets, permutation patterns",
    "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": "Rectangulations"
        },
        {
            "word": "combinatorial enumeration"
        },
        {
            "word": "generating functions"
        },
        {
            "word": "asymptotic enumeration"
        },
        {
            "word": "posets"
        },
        {
            "word": "permutation patterns"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/92s0c9qg",
    "frozenauthors": [
        {
            "first_name": "Andrei",
            "middle_name": "",
            "last_name": "Asinowski",
            "name_suffix": "",
            "institution": "Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria",
            "department": ""
        },
        {
            "first_name": "Jean",
            "middle_name": "",
            "last_name": "Cardinal",
            "name_suffix": "",
            "institution": "Département d’Informatique, Université libre de Bruxelles (ULB), 1050 Brussels, Belgium",
            "department": ""
        },
        {
            "first_name": "Stefan",
            "middle_name": "",
            "last_name": "Felsner",
            "name_suffix": "",
            "institution": "Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany",
            "department": ""
        },
        {
            "first_name": "Éric",
            "middle_name": "",
            "last_name": "Fusy",
            "name_suffix": "",
            "institution": "LIGM, CNRS, Université Gustave Eiffel, ESIEE Paris, 77454 Marne-la-Vallée, France",
            "department": ""
        }
    ],
    "date_submitted": "2025-03-14T16:58:21Z",
    "date_accepted": "2025-03-14T16:58:21Z",
    "date_published": "2025-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64974/galley/49784/download/"
        }
    ]
}