Partially Directed Snake Polyominoes

TL;DR

This paper investigates partially directed snake polyominoes across multiple dimensions, deriving generating functions and exploring their properties, including inscribed snakes and conjectures on maximal length configurations.

Contribution

It introduces functional equations and generating functions for partially directed snake polyominoes in various dimensions, advancing understanding of their combinatorial structure.

Findings

Derived length generating functions for 2D, 3D, and N-dimensional snakes.

Established two-variable generating functions for snakes inscribed in rectangles.

Formulated a conjecture on maximal length inscribed snake polyominoes.

Abstract

The goal of this paper is to study the family of snake polyominoes. More precisely, we focus our attention on the class of partially directed snakes. We establish functional equations and length generating functions of two dimensional, three dimensional and then $N$ dimensional partially directed snake polyominoes. We then turn our attention to partially directed snakes inscribed in a $b\times k$ rectangle and we establish two-variable generating functions, with respect to height $k$ and length $n$ of the snakes. We include observations on the relationship between snake polyominoes and self-avoiding walks. We conclude with a discussion on inscribed snakes polyominoes of maximal length which lead us to the formulation of a conjecture encountered in the course of our investigations.