{"pk":64878,"title":"Maximum entropy and integer partitions","subtitle":null,"abstract":"We derive asymptotic formulas for the number of integer partitions with given sums of \\(j\\)th powers of the parts for \\(j\\) belonging to a finite, non-empty set \\(J \\subset \\mathbb N\\). The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions. Finding the polynomial corrections and leading constant involves two steps: quantifying the error in approximating a discrete optimization problem by a continuous one and proving a multivariate local central limit theorem.\n \nMathematics Subject Classifications: 05A17, 05A16, 60F05\n \nKeywords: Integer partitions, maximum entropy, asymptotic enumeration, local central limit theorem, limit shape","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":"Integer partitions"},{"word":"maximum entropy"},{"word":"asymptotic enumeration"},{"word":"local central limit theorem"},{"word":"limit shape"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/0nw2d8hb","frozenauthors":[{"first_name":"Gweneth","middle_name":"","last_name":"McKinley","name_suffix":"","institution":"Department of Mathematics, University of California San Diego, California, U.S.A.","department":""},{"first_name":"Marcus","middle_name":"","last_name":"Michelen","name_suffix":"","institution":"Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Illinois, U.S.A.","department":""},{"first_name":"Will","middle_name":"","last_name":"Perkins","name_suffix":"","institution":"School of Computer Science, Georgia Institute of Technology, Georgia, U.S.A.","department":""}],"date_submitted":"2023-03-14T15:33:33Z","date_accepted":"2023-03-14T15:33:33Z","date_published":"2023-03-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64878/galley/49688/download/"}]}