Last Car Decomposition of Planar Maps

TL;DR

This paper introduces new equations for generating functions of planar maps with boundaries, inspired by parking tree decompositions, advancing combinatorial enumeration techniques.

Contribution

It applies a novel last car decomposition approach to peeling trees of planar maps, providing new characterizations for quadrangulations and triangulations.

Findings

Derived new equations for generating functions of planar maps.

Extended last car decomposition to peeling trees of planar maps.

Enhanced understanding of boundary conditions in planar map enumeration.

Abstract

We give new equations which characterize the generating functions of planar quadrangulations and planar triangulations, with zero, one or two boundaries. The proof is inspired by the Lackner--Panholzer last car decomposition of parking trees (arXiv:1504.04972) and consists in applying a similar decomposition to the peeling trees of planar maps.