{"pk":64899,"title":"Sub-Fibonacci behavior in numerical semigroup enumeration","subtitle":null,"abstract":"In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most \\(3\\) and that the number \\(n_g\\) of numerical semigroups of a genus \\(g\\) is asymptotic to \\(S\\varphi^g\\), where \\(S\\) is some positive constant and \\(\\varphi \\approx 1.61803\\) is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause \\(n_g\\) to deviate from a perfect exponential, including the number of semigroups with depth at least \\(4\\). Among other applications, these results imply the sharpest known asymptotic bounds on \\(n_g\\) and shed light on a conjecture by Bras-Amorós (2008) that \\(n_g \\geq n_{g-1} + n_{g-2}\\). Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.\n \nMathematics Subject Classifications: 20M14, 05A15, 05A16\n \nKeywords: Numerical semigroup, genus, Kunz coordinate, graph homomorphism","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":"Numerical semigroup"},{"word":"genus"},{"word":"Kunz coordinate"},{"word":"graph homomorphism"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/2qm1c11d","frozenauthors":[{"first_name":"Daniel","middle_name":"G.","last_name":"Zhu","name_suffix":"","institution":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.","department":""}],"date_submitted":"2023-09-14T07:41:21Z","date_accepted":"2023-09-14T07:41:21Z","date_published":"2023-09-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64899/galley/49709/download/"}]}