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{
    "pk": 64927,
    "title": "Decompositions of packed words and self duality of Word Quasisymmetric Functions",
    "subtitle": null,
    "abstract": "By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. However, the only known explicit isomorphism due to Vargas does not respect the bidendriform structure. This structure is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts vanish). In this paper, we construct two bases indexed by two new combinatorial families called red (dual side) and blue (primal side) biplane forests in bijection with packed words. In those bases, primitive elements are indexed by biplane trees and totally primitive elements by a certain subset of trees. We carefully combine red and blue forests to get bicolored forests. A simple recoloring of the edges allows us to obtain the first explicit bidendriform automorphism of WQSym.\n \nMathematics Subject Classifications: 05A05, 05A19, 05E05, 05E18\n \nKeywords: Bidendriform Hopf algebras, Word Quasisymmetric Functions, packed   words, permutation, primitive elements, duality, tree, forest, global descents",
    "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": "Bidendriform Hopf algebras"
        },
        {
            "word": "Word Quasisymmetric Functions"
        },
        {
            "word": "packed   words"
        },
        {
            "word": "permutation"
        },
        {
            "word": "primitive elements"
        },
        {
            "word": "duality"
        },
        {
            "word": "tree"
        },
        {
            "word": "forest"
        },
        {
            "word": "global descents"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/80g252tm",
    "frozenauthors": [
        {
            "first_name": "Hugo",
            "middle_name": "",
            "last_name": "Mlodecki",
            "name_suffix": "",
            "institution": "Université Paris-Saclay, Laboratoire Interdisciplinaire des Sciences du Numérique, Orsay, France",
            "department": ""
        }
    ],
    "date_submitted": "2024-06-28T15:19:53+02:00",
    "date_accepted": "2024-06-28T15:19:53+02:00",
    "date_published": "2024-06-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64927/galley/49737/download/"
        }
    ]
}