{"pk":64847,"title":"Monotone subsets in lattices and the Schensted shape of a Sós permutation","subtitle":null,"abstract":"For a fixed irrational number $\\alpha$ and $n\\in \\mathbb{N}$, we look at the shape of the sequence $(f(1),\\ldots,f(n))$ after Schensted insertion, where $f(i) = \\alpha i \\mod 1$. Our primary result is that the boundary of the Schensted shape is approximated by a piecewise linear function with at most two slopes. This piecewise linear function is explicitly described in terms of the continued fraction expansion for $\\alpha$. Our results generalize those of Boyd and Steele, who studied longest monotone subsequences. Our proofs are based on a careful analysis of monotone sets in two-dimensional lattices.\n \nMathematics Subject Classifications: 05A05, 11H06, 11B57, 11K06\n \nKeywords: Longest increasing subsequence, Schensted shape, geometry of numbers, S\\'os permutations","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":"Longest increasing subsequence"},{"word":"Schensted shape"},{"word":"geometry of numbers"},{"word":"S\\'os permutations"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/18b556r3","frozenauthors":[{"first_name":"Karl","middle_name":"","last_name":"Liechty","name_suffix":"","institution":"Department of Mathematical Sciences, DePaul University, Chicago, IL, U.S.A.","department":""},{"first_name":"T.","middle_name":"Kyle","last_name":"Petersen","name_suffix":"","institution":"Department of Mathematical Sciences, DePaul University, Chicago, IL, U.S.A.","department":""}],"date_submitted":"2022-06-25T19:29:47Z","date_accepted":"2022-06-25T19:29:47Z","date_published":"2022-06-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64847/galley/49657/download/"}]}