Enumeration of parallelogram polycubes

TL;DR

This paper develops enumeration formulas for parallelogram polycubes, linking them to multiple zeta functions and providing explicit generating functions based on geometric parameters, with extensions to higher dimensions.

Contribution

It introduces a novel enumeration approach for parallelogram polycubes using zeta functions and derives explicit formulas and generating functions, extending to polyhypercubes.

Findings

Explicit enumeration formulas for parallelogram polycubes

Connection between polycube enumeration and multiple zeta functions

Generalization to polyhypercubes

Abstract

In this paper, we enumerate parallelogram polycubes according to several parameters. After establishing a relation between Multiple Zeta Function and the Dirichlet generating function of parallelogram polyominoes, we generalize it to the case of parallelogram polycubes. We also give an explicit formula and an ordinary generating function of parallelogram polycubes according to the width, length and depth, by characterizing its projections. Then, these results are generalized to polyhypercubes.