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More models of walks avoiding a quadrant (extended abstract)

Mireille Bousquet-Melou, Michael Wallner

TL;DR

This work extends the enumeration of walks avoiding the negative quadrant to the king model with all eight neighbour steps, showing that the algebraicity phenomenon observed in prior non-convex-cone cases persists. The authors develop a robust framework based on an orbit-sum-zero auxiliary series, a quadrant-like reduction, and a generalized quadratic method, complemented by a guess-and-check computational approach to obtain explicit algebraic descriptions. They prove that the king model yields an algebraic $A(x,y)$ that yields $C(x,y)$ as a corrected combination of quadrant data, with the key boundary generating function $S(x)=t x M(0,x)$ satisfying a high-degree polynomial equation (degree 72 after depression), and they provide a full algebraic structure for the solution. The study also outlines a path to solve four additional models solvable via the reflection principle, while acknowledging substantial computational challenges due to very large-degree algebraic series. Overall, the paper advances the understanding of algebraicity in quadrant-avoidance problems and outlines a scalable strategy for more complex models, including explicit polynomial equations and asymptotic implications.

Abstract

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{é}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{é}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{é}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.

More models of walks avoiding a quadrant (extended abstract)

TL;DR

This work extends the enumeration of walks avoiding the negative quadrant to the king model with all eight neighbour steps, showing that the algebraicity phenomenon observed in prior non-convex-cone cases persists. The authors develop a robust framework based on an orbit-sum-zero auxiliary series, a quadrant-like reduction, and a generalized quadratic method, complemented by a guess-and-check computational approach to obtain explicit algebraic descriptions. They prove that the king model yields an algebraic that yields as a corrected combination of quadrant data, with the key boundary generating function satisfying a high-degree polynomial equation (degree 72 after depression), and they provide a full algebraic structure for the solution. The study also outlines a path to solve four additional models solvable via the reflection principle, while acknowledging substantial computational challenges due to very large-degree algebraic series. Overall, the paper advances the understanding of algebraicity in quadrant-avoidance problems and outlines a scalable strategy for more complex models, including explicit polynomial equations and asymptotic implications.

Abstract

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{é}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{é}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{é}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.

Paper Structure

This paper contains 15 sections, 3 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Enumeration in the three-quarter plane
  3. Interesting step sets
  4. A functional equation
  5. The king walks
  6. Reduction to an equation with orbit sum zero
  7. Reduction to a quadrant-like problem
  8. An equation between M(0,x), M(0,1/x), and M(x,0)
  9. An equation between M(0,x) and M(0,1/x)
  10. An equation for M(0,x) only
  11. The generalized quadratic method
  12. Guess-and-check
  13. Back to S(x) and R(x)
  14. More models
  15. Final polynomial equation for S(x) in the king model

Key Result

Theorem 1

Take the step set $\{ -1,0,1\}^2\setminus\{(0,0)\}$ and let $Q(x,y)$ be the generating function of lattice walks starting from $(0,0)$ that are confined to the first quadrant $\mathcal{Q}$ (this series is D-finite and given in bomi10). Then, the generating function of walks starting from $(0,0)$, co where $P(x,y)$ and $M(x,y)$ are algebraic of degree $216$ over ${\mathbb Q}(x,y,t)$. Of course, the

Figures (1)

  • Figure 1: The seven step sets to which the strategy of this paper should apply. The first two are solved in Bousquet2016, the third one in this paper.

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 2
  • Lemma 3: Non-negative part at pole $\rho$