Mountain Counting

On the Enumeration of Three-Dimensional Stacks of Spheres

  • Aaron Thomas University of Louisvile
  • Russell May Morehead State University
Keywords: enumerative combinatorics; generating functions


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.

How to Cite
Thomas, A., & May, R. (2022). Mountain Counting: On the Enumeration of Three-Dimensional Stacks of Spheres. The PUMP Journal of Undergraduate Research, 5, 70-79. Retrieved from