{"pk":64929,"title":"Intervals in the greedy Tamari posets","subtitle":null,"abstract":"We consider a greedy version of the \\(m\\)-Tamari order defined on \\(m\\)-Dyck paths, recently introduced by Dermenjian. Inspired by intriguing connections between intervals in the ordinary {1-}Tamari order and planar triangulations, and more generally by the existence of simple formulas counting intervals in the ordinary \\(m\\)-Tamari orders, we investigate the number of intervals in the greedy order on \\(m\\)-Dyck paths of fixed size. We find again a simple formula, which also counts certain planar maps (of prescribed size) called \\((m+1)\\)-constellations.\nFor instance, when \\(m=1\\) the number of intervals in the greedy order on {1-}Dyck paths of length \\(2n\\) is proved to be \\(\\frac{3\\cdot 2^{n-1}}{(n+1)(n+2)} \\binom{2n}{n}\\), which is also the number of bipartite maps with \\(n\\) edges.\nOur approach is recursive, and uses a \"catalytic\" parameter, namely the length of the final descent of the upper path of the interval. The resulting bivariate generating function is algebraic for all \\(m\\). We show that the same approach can be used to count intervals in the ordinary \\(m\\)-Tamari lattices as well. We thus recover the earlier result of Bousquet-Mélou, Fusy and Préville-Ratelle, who were using a different catalytic parameter.\n \nMathematics Subject Classifications: 05A15, 06A07, 06A11\n \nKeywords: Tamari posets, planar maps, enumeration, algebraic generating functions","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":"Tamari posets"},{"word":"planar maps"},{"word":"enumeration"},{"word":"algebraic generating functions"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/9342r6p5","frozenauthors":[{"first_name":"Mireille","middle_name":"","last_name":"Bousquet-Mélou","name_suffix":"","institution":"CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence Cedex, France","department":""},{"first_name":"Frédéric","middle_name":"","last_name":"Chapoton","name_suffix":"","institution":"IRMA, UMR 7501, Université de Strasbourg and CNRS, 7 rue René-Descartes, 67000 Strasbourg, France","department":""}],"date_submitted":"2024-07-01T09:30:09Z","date_accepted":"2024-07-01T09:30:09Z","date_published":"2024-06-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64929/galley/49739/download/"}]}