Walks avoiding a quadrant and the reflection principle

TL;DR

This paper advances the enumeration of lattice walks avoiding a quadrant by solving the king model, revealing the generating function as a sum of explicit D-finite and algebraic series, and extends algebraicity results to new step sets.

Contribution

It provides a detailed solution for the king model and proves algebraicity for three Weyl step sets, extending previous work on quadrant-avoiding walks.

Findings

The generating function is a sum of a D-finite and an algebraic series.

Proved algebraicity for three Weyl step sets.

Predicted algebraic structure for remaining step sets.

Abstract

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We expect a similar algebraicity phenomenon to hold for the seven Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.