{"pk":64958,"title":"Repeatable patterns and the maximum multiplicity of a generator in a reduced word","subtitle":null,"abstract":"We study the maximum multiplicity \\(\\mathcal{M}(k,n)\\) of a simple transposition \\(s_k=(k \\: k+1)\\) in a reduced word for the longest permutation \\(w_0=n \\: n-1 \\: \\cdots \\: 2 \\: 1\\), a problem closely related to much previous work on sorting networks and on the \"\\(k\\)-set\" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed \\(k\\) and sufficiently large \\(n\\), the optimal density is realized by paths which are periodic in a precise sense, so that \\[ \\mathcal{M}(k,n)=c_k n + p_k(n) \\] for a periodic function \\(p_k\\) and constant \\(c_k\\). In fact we show that \\(c_k\\) is always rational, and compute several bounds and exact values for this quantity with repeatable patterns, which we introduce.\n \nMathematics Subject Classifications: 05A05, 05A16, 05E99\n \nKeywords: Reduced words, permutations, \\(k\\)-sets, wiring diagram, weakly separated","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":"Reduced words"},{"word":"permutations"},{"word":"\\(k\\)-sets"},{"word":"wiring diagram"},{"word":"weakly separated"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/8vg0d5px","frozenauthors":[{"first_name":"Christian","middle_name":"","last_name":"Gaetz","name_suffix":"","institution":"Department of Mathematics, University of California, Berkeley, CA, U.S.A.","department":""},{"first_name":"Yibo","middle_name":"","last_name":"Gao","name_suffix":"","institution":"Beijing International Center for Mathematical Research, Peking University, Beijing, China","department":""},{"first_name":"Pakawut","middle_name":"","last_name":"Jiradilok","name_suffix":"","institution":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.","department":""},{"first_name":"Gleb","middle_name":"","last_name":"Nenashev","name_suffix":"","institution":"Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia","department":""},{"first_name":"Alexander","middle_name":"","last_name":"Postnikov","name_suffix":"","institution":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.","department":""}],"date_submitted":"2024-09-27T12:16:01+03:00","date_accepted":"2024-09-27T12:16:01+03:00","date_published":"2024-09-30T10:00:00+03:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64958/galley/49768/download/"}]}