Séminaire Combinatoire et Théorie des Nombres ICJ

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

by Bishal Deb (LPSM)

Salle Fokko du Cloux, Bât Braconnier (ICJ, Université Lyon 1)

Salle Fokko du Cloux, Bât Braconnier

ICJ, Université Lyon 1

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).

We will begin this talk by taking a look at some of these multivariate continued fractions. We will then sketch out the Foata--Zeilberger bijection. We will then introduce a Laguerre digraph which is a digraph in which each vertex has in- and out-degrees 0 or 1. We then provide a new interpretation of the Foata–Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. This interpretation enables us to prove some conjectured continued fractions due to Sokal and Zeng (2022) in the case of permutations, and Randrianarivony and Zeng (1996) and Deb and Sokal (2022 arXiv) in the case of D-permutations.