Exact solution of two classes of prudent polygons

TL;DR

This paper derives exact generating functions for two subclasses of prudent polygons, revealing that one is algebraic while the other is non-D-finite, advancing understanding of these self-avoiding walk models.

Contribution

It provides the first exact solutions for the generating functions of two classes of prudent polygons, highlighting their algebraic and non-D-finite nature.

Findings

Half-perimeter generating functions are algebraic and non-D-finite.

Provides explicit algebraic form for one subclass.

Establishes the non-D-finite nature of the other subclass.

Abstract

Prudent walks are self-avoiding walks on the square lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex is adjacent to the first one. More precisely, we give the half-perimeter generating functions of two subclasses of prudent polygons, which turn out to be algebraic and non-D-finite, respectively.