Plateau Polycubes and Lateral Area

TL;DR

This paper introduces enumeration formulas and generating functions for directed plateau polycubes and plateau polycubes based on width and lateral area, providing new asymptotic insights and linking to column-convex polyominoes.

Contribution

It presents explicit enumeration formulas and generating functions for two families of polycubes with respect to lateral area, a new parameter, and derives asymptotic results.

Findings

Explicit formulas for directed plateau polycubes and plateau polycubes.

Generating functions for these polycubes.

Asymptotic enumeration results.

Abstract

In this paper, we enumerate two families of polycubes, the directed plateau polycubes and the plateau polycubes, with respect to the width and a new parameter, the Lateral Area. We give an explicit formula and the generating function for each of the two families of polycubes. Moreover, some asymptotic results about plateau polycubes are provided. We also establish results concerning the enumeration of column-convex polyominoes that are useful to get asymptotic results of polycubes.