Enumeration of three-quadrant walks via invariants: some diagonally symmetric models

TL;DR

This paper addresses the enumeration of walks confined to the non-convex three-quadrant cone by extending Tutte-style invariants from quadrant models to a family of diagonally symmetric three-quadrant step sets. It develops a decoupling framework and constructs new invariant pairs that relate the three-quadrant generating function $C(x,y;t)$ to quadrant generating functions of a companion model, enabling uniform, model-by-model resolutions. The main contributions include algebraic solutions for the Kreweras-trilogy models, a D-algebraic solution for a sixth infinite-group model, and a D-finite solution for the diagonal model, with explicit expressions for endpoint generating functions and harmonic functions. Additional results cover the reverse and double Kreweras steps, as well as the simple and diagonal three-quadrant variants, all unified under the invariant methodology. The work clarifies when invariant-based methods yield exact, highly structured generating functions and provides explicit harmonic descriptions, offering a pathway to solving further three-quadrant models and informing potential extensions to less symmetric or more general step sets.

Abstract

In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of them deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps. More recently, similar questions have been raised for non-convex cones, typically the three-quadrant cone $C = \{ (i,j) : i \geq 0 \text{ or } j \geq 0 \}$. They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in $C$. This collection consists of diagonally symmetric models in $\{-1, 0,1\}^2\setminus\{(-1,1), (1,-1)\}$. Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte's notion of \emph{invariants}, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model $\{ \nearrow, \nwarrow, \swarrow, \searrow\}$, which is D-finite. The three algebraic models are those of the Kreweras trilogy, $S=\{\nearrow, \leftarrow, \downarrow\}$, $S'=\{\rightarrow, \uparrow, \swarrow\}$, and $S\cup S'$. Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in $S$ is an explicit rational function in the quadrant generating function with steps in $\{(j-i,j): (i,j) \in S\}$. We derive various corollaries, including an explicit algebraic description of the positive harmonic function in $C$ for the five models that are at least D-finite.

Figures (4)

• Figure 1: Two walks with Kreweras steps $\nearrow, \leftarrow, \downarrow$, one in the first quadrant $\mathcal{Q}$ (left), and one in three-quadrant cone $\mathcal{C}$ (right). The associated generating functions are algebraic.
• Figure 2: The series $D(xy)$ counts walks ending on the diagonal, and $\bar{x} U(\bar{x}, xy)$ those ending above the diagonal.
• Figure 3: The extension of $\mathbb{Q}(t)$ of degree $16$ generated by $A_1$. All elementary extensions have degree $2$. The series $C(1,1)$ has degree $16$ over $\mathbb{Q}(t)$, as $A_1$, but lies in a different extension of $\mathbb{Q}(t)$.
• Figure 4: Plots of the sequences \ref{['sequences']} against $1/n$, for $n\le 150$.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Definition 2.4