A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes

TL;DR

A novel bijection between stacked directed polyominoes and Motzkin paths with alternative catastrophes is presented and how this new connection can be used in order to obtain a better understanding of certain parameters of stacked directed animals is shown.

Abstract

We present a novel bijection between stacked directed polyominoes and Motzkin paths with alternative catastrophes. Further, we show how this new connection can be used in order to obtain a better understanding of certain parameters of stacked directed animals.

Figures (8)

• Figure 1: Different types of heaps of dimers.
• Figure 2: Constructing the connected heap $V(A)$ from an animal $A$ on the square grid.
• Figure 3: The schematic structure of stacked directed animals (left) and multi-directed animals (right) from Definitions \ref{['def:stacked_directed_animals']} and \ref{['def:multi_directed_animals']}, respectively LatticeAnimals. Each triangle represents a directed animal from Definition \ref{['def:directed_animal']}.
• Figure 4: Example of a Dyck excursion with alternative catastrophes marked in red; see Definition \ref{['def:catastrophealternative']}. The first and last catastrophe are not classical catastrophes in the sense of Definition \ref{['def:catastropheclassical']}. Note that a step from altitude $1$ to $0$ can either be an alternative catastrophe (red) or a step $-1$ from the step set $\mathcal{S}$ (black).
• Figure 5: The factorizations of half-pyramids and Motzkin excursions.

Theorems & Definitions (19)

• Definition 1.1: Lattice animals
• Definition 1.2: Directed animals
• Definition 1.3: Heaps of dimers
• Definition 1.4: Mapping from directed animals to heaps
• Remark 1.5
• Definition 1.6: Mapping from lattice animals to heaps
• Definition 1.7: Multi-directed animals
• Definition 1.8: Stacked directed animals
• Definition 2.1: Lattice paths
• Definition 2.2: Catastrophe