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{
    "pk": 64881,
    "title": "Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron",
    "subtitle": null,
    "abstract": "The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher's result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions for contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumerate bipartite planar trees and forests, and we apply our results to enumerate various families of permutations: for example, bipartite planar trees are in bijection with separable permutations. It is postulated by Livia Ferro, Tomasz Łukowski and Robert Moerman (2020) that contracted Grassmannian forests are in bijection with boundary strata of the momentum amplituhedron, an object encoding the tree-level S-matrix of maximally supersymmetric Yang-Mills theory. With this assumption, our results give a rank generating function for the boundary strata of the momentum amplituhedron, and imply that the Euler characteristic of the momentum amplituhedron is \\(1\\).\n \nMathematics Subject Classifications: 05A05, 05A15, 05C10\n \nKeywords: Generating functions, permutations, planar forests",
    "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": "Generating functions"
        },
        {
            "word": "permutations"
        },
        {
            "word": "planar forests"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/8xv6q5w3",
    "frozenauthors": [
        {
            "first_name": "Robert",
            "middle_name": "",
            "last_name": "Moerman",
            "name_suffix": "",
            "institution": "Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, U.K.",
            "department": ""
        },
        {
            "first_name": "Lauren",
            "middle_name": "K.",
            "last_name": "Williams",
            "name_suffix": "",
            "institution": "Department of Mathematics, Harvard University, Cambridge, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2023-03-14T15:53:35Z",
    "date_accepted": "2023-03-14T15:53:35Z",
    "date_published": "2023-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64881/galley/49691/download/"
        }
    ]
}