{"pk":64852,"title":"Counting lattice paths by crossings and major index I: the corner-flipping bijections","subtitle":null,"abstract":"We solve two problems regarding the enumeration of lattice paths in $\\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings. The first problem considers crossings of a single path with a fixed horizontal line. The second one counts pairs of paths with respect to the number of times they cross each other. Our proofs introduce lattice path bijections with convenient visual descriptions, and the answers are given by remarkably simple formulas involving $q$-binomial coefficients.\n \nMathematics Subject Classifications: 05A19, 05A15, 05A30\n \nKeywords: Lattice path, major index, crossing, valley, bijection","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":"Lattice path"},{"word":"major index"},{"word":"crossing"},{"word":"valley"},{"word":"bijection"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/58x811km","frozenauthors":[{"first_name":"Sergi","middle_name":"","last_name":"Elizalde","name_suffix":"","institution":"Department of Mathematics, Dartmouth College, Hanover, NH, U.S.A.","department":""}],"date_submitted":"2022-06-25T20:20:21Z","date_accepted":"2022-06-25T20:20:21Z","date_published":"2022-06-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64852/galley/49662/download/"}]}