Mountain Counting

On the Enumeration of Three-Dimensional Stacks of Spheres

Abstract

We investigate the combinatorial objects of mountains of spheres, which are three-dimensional variants of two-dimensional fountains of coins. A mountain can be decomposed into a sequence of fountains, and this decomposition leads to a recurrence relation. We obtain a closed-form solution for their counts based on this recurrence relation. Lastly, we discuss the computational feasibility of this enumeration.