3D Polyominoes inscribed in a rectangular prism

TL;DR

This paper introduces minimal 3D polyominoes inscribed in rectangular prisms, extending 2D concepts, and provides generating functions, formulas, and recurrences for their enumeration.

Contribution

It defines and analyzes minimal 3D polyominoes inscribed in rectangular prisms, deriving their generating functions and exact enumeration formulas.

Findings

Generated rational functions for minimal 3D polyominoes

Derived exact formulas and recurrences for specific sub-families

Extended 2D inscribed polyomino concepts to 3D

Abstract

We introduce a family of 3D combinatorial objects that we define as minimal 3D polyominoes inscribed in a rectanglar prism. These objects are connected sets of unitary cubic cells inscribed in a given rectangular prism and of minimal volume under this condition. They extend the concept of 2D polyominoes inscribed in a rectangle defined in a previous work. Using their geometric structure and elementary combinatorial arguments, we construct generating functions of minimal 3D polyominoes in the form of rational functions. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes.