Enumeration of Boundary Cubic Inner-Forest Maps

Abstract

This paper investigates the number of boundary cubic inner-forest maps and presents some formulae for such maps with the size (number of edges) and the valency of the root-face as two parameters. Further, by duality, some corresponding results for rooted outer-planar maps are obtained. It is also an answer to the open problem in \([15]\) and corrects the result on boundary cubic inner-tree maps in \([15]\).