CONSTELLATIONS AND MULTICONTINUED FRACTIONS: APPLICATION TO EULERIAN TRIANGULATIONS

Niveau: Supérieur
CONSTELLATIONS AND MULTICONTINUED FRACTIONS: APPLICATION TO EULERIAN TRIANGULATIONS MARIE ALBENQUE1 AND JEREMIE BOUTTIER2 Abstract. We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determi- nants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance. 1. Introduction From the initial work of Hurwitz (1891) about transitive ordered factorizations of the identity in the symmetric group, to the bijective approach of Bousquet-Melou and Schaeffer (2000) and Bouttier et al. (2004), including the more algebraic approach of Goulden and Jackson (1997), constellations appear in many forms in different areas of combinatorics. We refer the reader to the book of Lando and Zvonkin (2004) for an extensive review of the variety of contexts in which constellations appear. In this paper, we focus on the map point of view, and consider the problem of enumerating planar constellations with two points at a prescribed distance. In Bouttier et al. (2004), this problem has already been tackled by giving a bijection between this family of constellations and a family of decorated trees called mobiles, which results in recurrence equations that characterize the associate generating functions.

• rises p1


• constellation generating


• root edge


• p1 yiqp1


• colored black


• np1 np1


• very-well-labeled trees


• functions


• black faces