{"pk":64969,"title":"Continued fractions using a Laguerre digraph interpretation of the Foata-Zeilberger bijection and its variants","subtitle":null,"abstract":"In the combinatorial theory of continued fractions, the Foata-Zeilberger bijection and its variants have been extensively used to derive various continued fractions enumerating several (sometimes infinitely many) simultaneous statistics on permutations (combinatorial model for factorials) and D-permutations (combinatorial model for Genocchi and median Genocchi numbers). A Laguerre digraph is a digraph in which each vertex has in- and out-degrees \\(0\\) or \\(1\\). In this paper, we interpret the Foata-Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. Using this interpretation, we obtain Jacobi-type continued fractions for multivariate polynomials enumerating permutations, and also Thron-type and Stieltjes-type continued fractions for multivariate polynomials enumerating D-permutations, in both cases including the counting of cycles. This enables us to prove some conjectured continued fractions due to Sokal and Zeng (2022 Advances in Applied Mathematics) in the case of permutations, and Randrianarivony and Zeng (1996 Electronic Journal of Combinatorics) and Deb and Sokal (2024 Advances in Applied Mathematics) in the case of D-permutations.\n \nMathematics Subject Classifications: 05A19 (Primary); 05A05, 05A10, 05A15, 05A30, 11B68, 30B70 (Secondary).\n \nKeywords: Permutations, D-permutations, continued fraction, Foata-Zeilberger bijection, S-fraction, J-fraction, T-fraction, Dyck path, almost-Dyck path, Motzkin path, Schröder path, Laguerre digraphs","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":"Permutations"},{"word":"D-permutations"},{"word":"continued fraction"},{"word":"Foata-Zeilberger bijection"},{"word":"S-fraction"},{"word":"J-fraction"},{"word":"T-fraction"},{"word":"Dyck path"},{"word":"almost-Dyck path"},{"word":"Motzkin path"},{"word":"Schröder path"},{"word":"Laguerre digraphs"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/35r6q8zv","frozenauthors":[{"first_name":"Bishal","middle_name":"","last_name":"Deb","name_suffix":"","institution":"Department of Mathematics, University College London, London WC1E 6BT, U.K.\nSorbonne Université and Université Paris Cité, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, 75005 Paris, France","department":""}],"date_submitted":"2025-03-15T01:42:12+09:00","date_accepted":"2025-03-15T01:42:12+09:00","date_published":"2025-03-15T16:00:00+09:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64969/galley/49779/download/"}]}