Stacked directed animals and multi-directed animals defined without using heaps of pieces

TL;DR

This paper introduces alternative definitions for stacked directed animals and multi-directed animals that do not rely on heaps of pieces, while maintaining equivalence with the original models.

Contribution

It provides new, heap-free definitions for these lattice models, simplifying their conceptual understanding without losing their original properties.

Findings

Definitions are equivalent to original models

Heaps of pieces are no longer required

Simplifies the conceptual framework of the models

Abstract

Stacked directed animals and multi-directed animals are two lattice models defined by Bousquet-M\'{e}lou and Rechnitzer in 2002. The original definitions of those models involve heaps of pieces, i.e., some geometric representation of partially commutative monoids. The object of this writing is to define stacked directed animals and multi-directed animals in such a way that heaps of pieces are not involved. Our alternative definitions are equivalent to Bousquet-M\'{e}lou and Rechnitzer's original ones.