Enumerating planar stuffed maps as hypertrees of mobiles

TL;DR

This paper introduces a bijection between bipartite planar stuffed maps and hypermobiles, generalizing known map-bijections, and derives algebraic and functional equations for their generating functions, enabling enumeration of complex stuffed quadrangulations.

Contribution

It establishes a new bijection between stuffed maps and hypertrees of mobiles, extending classical map bijections and deriving associated generating function equations.

Findings

Derived algebraic and functional equations for generating functions.

Explicit enumeration of stuffed quadrangulations.

Generalization of classical map bijections.

Abstract

A planar stuffed map is an embedding of a graph into the 2-sphere $S^{2}$, considered up to orientation-preserving homeomorphisms, such that the complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to $S^{2}$ with multiple boundaries. This is a generalization of planar maps whose complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to a disc. The main goal of this work is to construct a bijection between bipartite planar stuffed maps and collections of integer-labelled trees connected by hyperedges such that they form a hypertree, called hypermobiles. This bijection directly generalizes the Bouttier-Di Franceso-Guitter bijection between bipartite planar maps and mobiles. Additionally, we show that the generating functions of these trees of mobiles satisfy both an algebraic equation, generalizing the case of ordinary planar maps, and a new functional equation. As an example, we explicitly enumerate a class of stuffed quadrangulations.