Constellations and multicontinued fractions: application to Eulerian triangulations

TL;DR

This paper develops a combinatorial approach using multicontinued fractions and Hankel determinants to enumerate planar constellations with two points at a fixed distance, leading to new insights on Eulerian triangulations.

Contribution

It introduces a novel algebraic framework linking constellations and multicontinued fractions for enumeration problems.

Findings

Derived the generating function for Eulerian triangulations with two fixed points.

Established a combinatorial correspondence between different families of constellations.

Applied algebraic methods to solve a classical enumeration problem.

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 determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance.