# Quarter-plane lattice paths with interacting boundaries: the Kreweras and reverse Kreweras models

## TL;DR

This paper analyzes how boundary interactions affect the algebraic and D-finite nature of generating functions in Kreweras and reverse Kreweras lattice path models, revealing new algebraic properties.

## Contribution

It provides a complete algebraic solution for boundary-interacting Kreweras and reverse Kreweras models using the algebraic kernel method, highlighting novel algebraic and D-finite cases.

## Key findings

- Reverse Kreweras models have algebraic generating functions for all boundary weights.
- Kreweras models have algebraic generating functions only when returning to the origin.
- This is the first example of a quarter-plane model with such properties.

## Abstract

Lattice paths in the quarter plane have led to a large and varied set of results in recent years. One major project has been the classification of step sets according to the properties of the corresponding generating functions, and this has involved a variety of techniques, some highly intricate and specialised. The famous Kreweras and reverse Kreweras walk models are two particularly interesting models, as they are among the only four cases which have algebraic generating functions. Here we investigate how the properties of the Kreweras and reverse Kreweras models change when boundary interactions are introduced. That is, we associate three real-valued weights $a,b,c$ with visits by the walks to the $x$-axis, the $y$-axis and the origin $(0,0)$ respectively. These models were partially solved in a recent paper by Beaton, Owczarek and Rechnitzer (2019). We apply the algebraic kernel method to completely solve these two models. We find that reverse Kreweras walks have an algebraic generating function for all $a,b,c$, regardless of whether the walks are restricted to end at the origin or on one of the axes, or may end anywhere at all. For Kreweras walks, the generating function for walks returning to the origin is algebraic, but the other cases are only D-finite. To our knowledge this is the first example of a quarter-plane model with this property.