{"pk":65008,"title":"Boolean elements in the Bruhat order","subtitle":null,"abstract":"We show that a Weyl group element is boolean if and only if it avoids a set of Billey-Postnikov patterns, which we describe explicitly. Our proof is based on analysis of inversion sets, and it is in large part type-uniform. We also introduce the notion of linear pattern avoidance, and show that boolean elements are characterized by avoiding just \\(3\\) linear patterns in types \\(A_2\\), \\(A_3\\), and \\(D_4\\), respectively.\nWe also consider the more general case of \\(k\\)-boolean Weyl group elements. We say that a Weyl group element \\(w\\) is \\(k\\)-boolean if every reduced expression for \\(w\\) contains at most \\(k\\) copies of each generator. We show that the \\(2\\)-boolean elements of the symmetric group are characterized by avoiding the patterns \\(3421,4312,4321,\\) and \\(456123\\), and obtain their generating function.\n \nMathematics Subject Classifications: 05A05, 20F55\n \nKeywords: Boolean permutations, Bruhat orders, Billey-Postnikov patterns, Weyl groups","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":"Boolean permutations"},{"word":"Bruhat orders"},{"word":"Billey-Postnikov patterns"},{"word":"Weyl groups"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/6v0196p6","frozenauthors":[{"first_name":"Yibo","middle_name":"","last_name":"Gao","name_suffix":"","institution":"Beijing International Center for Mathematical Research, Peking University, Beijing, China","department":""},{"first_name":"Kaarel","middle_name":"","last_name":"Hänni","name_suffix":"","institution":"The Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, U.S.A.","department":""}],"date_submitted":"2025-09-12T12:12:40Z","date_accepted":"2025-09-12T12:12:40Z","date_published":"2025-09-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/65008/galley/49818/download/"}]}