Directed and multi-directed animals on the king's lattice

TL;DR

This paper introduces the directed king's lattice, a new solvable lattice for directed animals, analyzes their generating functions, extends to multi-directed animals, and develops efficient sampling algorithms.

Contribution

It presents the directed king's lattice, derives algebraic and non-D-finite generating functions for animals, and introduces sampling algorithms.

Findings

Directed animals have algebraic generating functions linked to Schr"oder numbers.

Multi-directed animals form a superclass with non-D-finite generating functions.

Efficient random sampling algorithms are proposed for these animals.

Abstract

This article introduces a new, simple solvable lattice for directed animals: the directed king's lattice, or square lattice with next nearest neighbor bonds and preferred directions {W, NW, N, NE, E}. We show that the directed animals in this lattice have an algebraic generating function linked to the Schr\"oder numbers and belong to the same universality class as the ones in the square and triangular lattices. We also define multi-directed animals in the king's lattice, which form a superclass of directed animals. We compute their generating function and show that it is not D-finite. Finally, we propose efficient random sampling algorithms for our animals.