Average site perimeter of directed animals on the two-dimensional lattices

TL;DR

This paper develops new combinatorial methods to compute the average site perimeter of directed animals on 2D lattices, confirming several conjectures and extending results across different lattice types.

Contribution

It introduces bijective combinatorial techniques based on Viennot's correspondence, enabling precise calculations of parameters like site perimeter for directed animals.

Findings

Computed average site perimeter for directed animals on square and triangular lattices.

Proved conjectures by Conway and Le Borgne regarding directed animals.

Extended combinatorial methods to bounded variants of lattices.

Abstract

We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source on the square lattice and its bounded variants, and we give counterparts for most of them in the triangular lattices. We thus prove conjectures by Conway and Le Borgne. The techniques used are based on Viennot's correspondence between directed animals and heaps of pieces (or elements of a partially commutative monoid).