Skew 2-Dyck Paths: Kernel Method Analysis

• Skew 2-Dyck paths are lattice walks defined by U, D, and L steps that avoid immediate UL and LU patterns, ensuring unique combinatorial behavior.
• The kernel method is applied to derive functional equations and an algebraic generating function that captures the enumeration and asymptotic growth of these paths.
• An automaton model and structured recurrences provide insights into state transitions and forbidden patterns, with extensions to generalized skew t-Dyck paths.

A skew 2-Dyck path is a lattice walk in $\mathbb{Z}^2$ of length $3n$, starting at $(0,0)$, never traversing below the $x$-axis, and composed of three types of steps: $U=(1,+1)$ (up), $D=(2,-2)$ (down), and $L=(2,-2)$ (left-down in "red," i.e., distinguished syntactically from $D$). The local forbidden patterns $UL$ and $LU$ (no up step immediately followed by a left-down, and vice versa) are enforced. Enumeration and structural results for these paths have recently been systematized via the kernel method, providing sharp algebraic and asymptotic statements as well as automata for pattern avoidance (Prodinger, 14 Dec 2025).

1. Combinatorial Definition and Local Constraints

A skew 2-Dyck path consists of steps $U=(1,+1)$, $D=(2,-2)$, and $L=(2,-2)$, with $L$ being notationally distinct from $D$. The walk starts at $(0,0)$, remains weakly above the $x$-axis, and never contains the immediate step sequences $UL$ or $LU$. This syntactic restriction ensures specific path behaviors, crucial for distinguishing skew 2-Dyck paths from ordinary Dyck or Motzkin-type paths.

In this setting, $D$ and $L$ have identical geometric vectors but constitute separate syntactic tokens. The length of a path is $3n$ to maintain the endpoint constraint at height zero: up-steps add 1, both $D$ and $L$ decrease height by 2, and forward displacements always total $3n$.

2. Automaton and State Model

The set of skew 2-Dyck paths can be captured by a finite automaton that records both the current height and the class of the last step taken. The automaton is trilevel, reflecting the last step being $U$, $D$, or $L$:

• Layer 1: Arrived by $U$; from here, only $U$ or $D$ steps are allowed.
• Layer 2: Arrived by $D$; all three steps are permitted.
• Layer 3: Arrived by $L$; only $D$ or $L$ are allowed (no $U$).

Every transition adjusts height: $U$ increases by $+1$; $D$ and $L$ decrease by $-2$. This automaton ensures that $UL$ and $LU$ are globally avoided. The originating state at $(0,0)$ is in Layer 1, and valid paths of interest return to height $0$ in any layer.

3. Functional Equations and Generating Functions

Let $f_i(z)$, $g_i(z)$, $h_i(z)$ denote the generating functions for prefixes ending at height $i$ in layers corresponding to $U$, $D$, $L$, respectively, with $z$ marking steps. The generating functions $F(u,z)$, $G(u,z)$, $H(u,z)$ are then

$$F(u,z) = \sum_{i\geq0} f_i(z)u^i,\quad G(u,z) = \sum_{i\geq0} g_i(z)u^i,\quad H(u,z) = \sum_{i\geq0} h_i(z)u^i.$$

These satisfy the recurrences:

• $f_0 = 1$, $f_{i+1} = z f_i + z g_i$
• $g_i = z f_{i+2} + z g_{i+2} + z h_{i+2}$
• $h_i = z g_{i+2} + z h_{i+2}$

Translating to functional equations:

• $(I)\ \ F(u,z) - 1 = z u [F(u,z) + G(u,z)]$
• $$(II)\ \ u^2G(u,z) = z[F(u,z) - f_0 - u f_1] + z[G(u,z) - g_0 - u g_1] + z[H(u,z) - h_0 - u h_1]$$
• $$(III)\ \ u^2 H(u,z) = z[G(u,z) - g_0 - u g_1] + z[H(u,z) - h_0 - u h_1]$$

These coupled equations form the basis for applying the kernel method (Prodinger, 14 Dec 2025).

4. Kernel Method Solution

Application of the kernel method yields that $F$, $G$, $H$ can each be written with a denominator

$$\text{Denominator}(u) = z u^4 - u^3 - z^2 u^2 + 2z u - z^3$$

with numerator and denominator sharing the roots $u - s_j(z)$. The cancellation of the "bad" branches—roots not expandable as power series in $z$—ensures that only the root $s_4(z)$ (the small power series branch) leads to formal power series solutions. After appropriate residue calculations and matching initial conditions,

$A(z) = 1 + g_0(z) + h_0(z)$

where $g_0(z), h_0(z)$ are determined in closed form:

$$g_0(z) = \frac{1}{z s_4(z)} - 1, \qquad h_0(z) = \frac{1}{z s_4(z)} - \frac{z}{s_4(z)^2} - 1.$$

The final algebraic generating function is

$$A(z) = \frac{1}{6z} + \frac{1}{3} - \frac{\sqrt{1 - 8z + 4z^2}}{6z}$$

which enumerates skew 2-Dyck paths of total length $3n$ (Prodinger, 14 Dec 2025).

5. Explicit Coefficient Formulas, Recurrence, and Asymptotics

The central enumerative result is the coefficient formula

$$a_n = [z^n]A(z) = \frac{1}{n} \sum_{i=0}^{n-1} 3^i \binom{n}{i} \binom{n}{i+1},\qquad n\ge 1, \ a_0=1$$

which provides integer counts of skew 2-Dyck paths of length $3n$.

A compact second-order recurrence is

$$(n+2)a_{n+2} = (8n+12)a_{n+1} - 4(2n+1)a_n, \qquad a_0=1, \ a_1=1.$$

Asymptotically,

$a_n \sim C\,\rho^{-n} n^{-3/2}$

where $\rho = \frac{2-\sqrt{2}}{4}$ is the dominant singularity, and $C \approx 1.25$ (Prodinger, 14 Dec 2025).

6. Generalization to Skew t-Dyck Paths

The analysis extends to "skew $t$-Dyck paths" (where $D$ and $L$ become $(t,-t)$ steps), leading to functional equations with $u^t$-shifts and a denominator of degree $2t$:

$$\Delta_t(u) = z u^{2t} - u^{2t-1} - z^2 u^{2t-2} + 2z u^t - z^3$$

Unique algebraic generating functions of degree $2t$ enumerate skew $t$-Dyck paths, recovering the $t=2$ case above, and delivering similar singularity/bijection phenomena (Prodinger, 14 Dec 2025).

7. Connections to Related Models and Methodologies

Comparable models have been studied for "decorated-Dyck" and "partial skew Dyck" paths, e.g., with color-distinguished down-steps or variants that allow additional southwest steps $(-1,-1)$ (subject to non-intersection constraints). The methodology—writing layered state recursions, translating to matrix-functional generating functions, and applying the kernel method—remains central and allows for the explicit enumeration of excursions, partial paths, and variations such as negative territory extension or step-weighted refinements (Prodinger, 2021).

The kernel method emerges as a powerful tool for resolving the algebraic structure of generating functions arising in these constrained lattice path models, providing residue-style cancellation of unwanted roots and yielding both exact and asymptotic enumerative formulae. Generalizations are systematic for other values of $t$ or for symmetric/dual models with interchange of up/down step structure.

These results bridge to classical problems in planar lattice path enumeration, algebraic generating functions, and k-ary tree bijections, and are of interest in combinatorics, statistical mechanics, and related fields.