paper

Continued fractions using a Laguerre digraph interpretation of the Foata--Zeilberger bijection and its variants

arXiv:2304.14487

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 or . 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.

arXiv commentLaTeX2e, 85 pages, 13 figures. v2: Added two running examples for illustration and 13 figures. Added several remarks based on comments from reviewers. To appear in Combinatorial Theory. arXiv admin note: text overlap with arXiv:2212.07232

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