On symmetric quadrangulations and triangulations

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.