More models of walks avoiding a quadrant (extended abstract)

TL;DR

This paper extends the enumeration of lattice paths avoiding a quadrant by analyzing king walks with all 8 directions, revealing complex algebraic generating functions and suggesting the algebraicity phenomenon persists in related models.

Contribution

It introduces a detailed solution for king walks avoiding a quadrant, demonstrating the algebraic nature of their generating functions and addressing larger algebraic series complexities.

Findings

Generated functions are algebraic but not D-finite.

The approach generalizes previous models with similar algebraic properties.

The algebraicity phenomenon likely persists in related quadrant models.

Abstract

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{\'e}lou, 2016]. We solve in detail a new case, the king walks, where all 8 nearest neighbour steps are allowed. As in the two cases solved in [Bousquet-M{\'e}lou, 2016], the associated generating function is proved to differ from a simple, explicit D-finite series (related to the enumeration of walks confined to the first quadrant) by an algebraic one. The principle of the approach is the same as in [Bousquet-M{\'e}lou, 2016], but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We also explain why we expect the observed algebraicity phenomenon to persist for 4 more models, for which the quadrant problem is solvable using the reflection principle.