Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones

TL;DR

This paper generalizes the enumeration of small-step walks confined to various M-quadrant cones, deriving functional equations and classifying the nature of their generating functions for any positive integer M.

Contribution

It introduces a unified analytic framework for enumerating walks on M-quadrant cones, extending known results from quarter-plane walks to more complex cone geometries.

Findings

Derived an analytic functional equation for M-quadrant walks.

Provided exact integral solutions for walks on a three-quadrant cone.

Classified the generating functions as algebraic, D-finite, or D-algebraic based on M and starting point.

Abstract

We address the enumeration of walks with small steps confined to a two-dimensional cone, for example the quarter plane, three-quarter plane or the slit plane. In the quarter plane case, the solutions for unweighted step-sets are already well understood, in the sense that it is known precisely for which cases the generating function is algebraic, D-finite or D-algebraic, and exact integral expressions are known in all cases. We derive similar results in a much more general setting: we enumerate walks on an $M$-quadrant cone for any positive integer $M$, with weighted steps starting at any point. The main breakthrough in this work is the derivation of an analytic functional equation which characterises the generating function of these walks, which is analogous to one now used widely for quarter-plane walks. In the case $M=3$, which corresponds to walks avoiding a quadrant, we provide exact integral-expression solutions for walks with weighted small steps which determine the generating function ${\sf C}(x,y;t)$ counting these walks. Moreover, for each step-set and starting point of the walk we determine whether the generating function ${\sf C}(x,y;t)$ is algebraic, D-finite or D-algebraic as a function of $x$ and $y$. In fact we provide results of this type for any $M$-quadrant cone, showing that this nature is the same for any odd $M$. For $M$ even we find that the generating functions counting these walks are D-finite in $x$ and $y$, and algebraic if and only if the starting point of the walk is on the same axis as the boundaries of the cone.