On symmetric quadrangulations and triangulations

TL;DR

This paper introduces new methods for enumerating symmetric planar maps, specifically quadrangulations and triangulations, and computes related series with boundary distance control using quotient operations.

Contribution

It presents novel enumeration techniques for symmetric simple quadrangulations and triangulations based on quotient operations, extending to polygon dissections with boundary distance analysis.

Findings

New enumeration formulas for symmetric quadrangulations and triangulations

Series computation for symmetric polygon dissections with boundary distance control

Application of quotient and substitution operations in map enumeration

Abstract

This article presents new enumerative results related to symmetric planar maps. In the first part a new way of enumerating rooted simple quadrangulations and rooted simple triangulations is presented, based on the description of two different quotient operations on symmetric simple quadrangulations and triangulations. In the second part, based on results of Bouttier, Di Francesco and Guitter and on quotient and substitution operations, the series of three families of symmetric quadrangular and triangular dissections of polygons are computed, with control on the distance from the central vertex to the outer boundary.