On Counting Rooted Triangular Maps

Abstract

Let R be a simply connected closed region in the Euclidean plane E2 whose boundary is a simple closed curve C. A triangular map, or simply "map," is a representation of R as the union of a finite number of disjoint point sets called cells, where the cells are of three kinds, vertices, edges, and faces (said to be of dimension 0, 1, and 2, respectively), where each vertex is a single point, each edge is an open arc whose ends are distinct vertices, and each face is a simply connected open region whose boundary consists of the closure of the union of three edges. Two cells of different dimension are incident if one is contained in the boundary of the other.