Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof

TL;DR

This paper provides a new purely combinatorial method for counting planar maps with specific vertex degrees, building on and generalizing previous matrix model solutions and recent combinatorial techniques.

Contribution

It introduces an alternative combinatorial approach to enumerate planar maps, extending existing methods to handle arbitrary vertex degrees.

Findings

Developed a generalized combinatorial framework for planar map enumeration

Provided an explicit counting method for maps with prescribed vertex degrees

Connected matrix model solutions with combinatorial tree techniques

Abstract

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.