{"pk":64823,"title":"Three Fuss-Catalan posets in interaction and their associative algebras","subtitle":null,"abstract":"We introduce $\\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $\\delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak Bruhat order of permutations, we construct algebras whose products form intervals of the lattices of $\\delta$-cliff. We provide necessary and sufficient conditions on $\\delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $\\delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra.\n \nMathematics Subject Classifications: 05E99, 06A07, 06A11, 16T30","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":"Poset"},{"word":"Tamari lattice"},{"word":"Fuss-Catalan objets"},{"word":"Malvenuto-Reutenauer algebra"},{"word":"Loday-Ronco algebra"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/4s55d1j5","frozenauthors":[{"first_name":"Camille","middle_name":"","last_name":"Combe","name_suffix":"","institution":"Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg et CNRS, Strasbourg, France","department":""},{"first_name":"Samuele","middle_name":"","last_name":"Giraudo","name_suffix":"","institution":"LIGM, Université Gustave Eiffel, CNRS, ESIEE Paris, Marne-la-Vallée, France","department":""}],"date_submitted":"2022-03-23T20:01:45Z","date_accepted":"2022-03-23T20:01:45Z","date_published":"2022-03-31T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64823/galley/49633/download/"}]}