Colored Motzkin Paths in Combinatorics

• Colored Motzkin paths are lattice paths with colored up, down, and horizontal steps that never fall below the x-axis, extending classical Motzkin paths.
• The article surveys enumeration methods using generating functions, continued fractions, and bijections that link these paths to tableaux, poset intervals, and permutation statistics.
• Advanced applications connect colored Motzkin paths to models in statistical physics, refined combinatorial identities, and asymptotic analysis of lattice structures.

Colored Motzkin paths are a rich generalization of classical Motzkin paths, introducing additional combinatorial parameters through the coloring or weighting of steps. These structures find deep connections with the enumeration of poset intervals, partition identities, permutation statistics, generalized trees, and more. At a foundational level, a colored Motzkin path of length $n$ is a restricted lattice path starting at $(0,0)$, ending at $(n,0)$, never passing below the $x$-axis, and consisting of up, down, and horizontal steps, each potentially distinguished by color, type, or position-dependent weight. This article surveys formal definitions, enumerative machinery, bijective correspondences, and advanced applications, focusing on colored Motzkin paths as they arise in arXiv research.

1. Definitions and Generalizations

Colored Motzkin paths extend classical Motzkin paths by assigning multiple colors or types to horizontal steps, and sometimes to up or down steps:

• Bicolored Motzkin paths: The horizontal step is split into two distinguishable types ($H_1$, $H_2$), so each step is $U$, $D$, $H_1$, or $H_2$ (Elizalde et al., 20 Dec 2024, Krattenthaler, 15 Jan 2025). In context, the coloring may correspond to refined combinatorial objects or encoding additional data.
• $(k,\ell)$-Motzkin paths: Motzkin paths where each horizontal step may be of $k$ colors, and each down step of $\ell$ colors. For example, (3,2)-Motzkin paths use three horizontal colors and two down colors (Chen et al., 2010).
• d-colored Motzkin paths ("Editor’s term"): Each up step and each down step receives a color label indexed $1\leq i\leq d$, and the level steps can be colored or uncolored, with nesting constraints on the colored parentheses (Eu et al., 2013).
• Order-$\ell$ colored Motzkin paths: Steps are $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ ($1\leq i\leq \ell$); color vectors $\vec\alpha$, $\vec\beta$ prescribe coloring schemes for steps landing at $y=0$ or $y>0$ (DeJager et al., 2020).
• Generalized/k-Fibonacci colored Motzkin paths: Allow horizontal steps $H_\ell=(\ell,0)$ of any length $\ell\geq1$, each colored according to $F_{k,\ell}$, the $k$-Fibonacci number (Castro et al., 2013).

These colorings encode refined parameters, e.g., set-valued tableau statistics, edge multiplicities in trees (Prodinger, 2021), or permutations with refined cycle/excedance structure (Elizalde, 2017).

2. Enumerative and Generating Function Frameworks

Colored Motzkin paths are enumerated via sophisticated generating functions, often quadratic or continued fractions:

• Standard functional equation:

For bicolored paths, the generating function $C(x,y)$ counting paths by $x$ for each $U$ or $H_1$ and $y$ for each $D$ or $H_2$ satisfies

$C(x,y) = 1 + (x+y)C(x,y) + x y C(x,y)^2,$

solved as

$C(x,y) = \frac{1-x-y-\sqrt{(1-x-y)^2-4xy}}{2xy}.$

Refined normalization yields

$$A(x,y) = \frac{2}{1 - x - y + 2xy + \sqrt{(1 - x - y)^2 - 4xy}},$$

which enumerates interval-closed sets for the product of chains (Elizalde et al., 20 Dec 2024).

• Multicolored cases:

For $(u,\ell,d)$-colored paths, the closed form is

$$G(z) = \frac{1 - Lz - \sqrt{(1 - Lz)^2 - 4UD z^2}}{2UD z^2}$$

where $U$, $L$, $D$ are total up, level, down weights, or sums of colors (Woelki, 2013).

• k-Fibonacci colored paths:

The generating function is

$$T_k(z) = \frac{1 - (k+1)z - z^2 - \sqrt{(1-(k+1)z-z^2)^2-4z^2(1-kz-z^2)^2}}{2z^2(1-kz-z^2)}$$

(Castro et al., 2013).

• Continued fractions:

For colored Motzkin paths arising from permutation statistics, Jacobi-type (J-) continued fractions encode weights per height:

$$J_{d,\ell}(z) = \cfrac{1}{1 - \ell_0 z - \cfrac{d_1 z^2}{1 - \ell_1 z - \cfrac{d_2 z^2}{\cdots}} }$$

(Elizalde, 2017).

• Refined enumeration with Lagrange inversion:

For two-colored paths encoding tableau statistics, Lagrange inversion yields explicit coefficient formulas (Krattenthaler, 15 Jan 2025).

3. Combinatorial and Bijective Correspondences

Colored Motzkin paths allow explicit bijections to refined combinatorial families:

• Interval-closed sets: Every interval-closed subset of $[m] \times [n]$ maps to a bicolored Motzkin path, with a normalization rule forbidding $H_2H_1$ at height 0 to prevent double-counting (Elizalde et al., 20 Dec 2024).
• Set-valued tableaux: Two-colored paths track row statistics; bijections map tableaux of 2-rowed shapes to colored paths, respecting the numbers of entries per row (Krattenthaler, 15 Jan 2025).
• Standard Young tableaux: $d$-Motzkin paths are in bijection with SYT of shape with height at most $2d+1$ (Eu et al., 2013).
• Noncrossing linked partitions: Large $(3,2)$-Motzkin paths correspond to noncrossing linked partitions; the two colors on level steps on the axis encode linked/non-linked blocks (Chen et al., 2010).
• Multi-edge trees: 3-colored Motzkin paths correspond to rooted ordered trees, where blue steps encode edge multiplicities (Prodinger, 2021).
• Permutation cycle diagrams: The cycle diagram mapping encodes permutation statistics as colored Motzkin path weights—down steps and level steps at height $h$ carry $h^2$ and $2h+1$ colors, respectively (Elizalde, 2017).
• Bijection to quarter-plane walks: Bicolored Motzkin paths of length $n$ correspond bijectively to quarter-plane walks in the first quadrant with six step types (Yeats, 2014). Variants (triangular domain, amplitude bounds) yield further bijections (Courtiel et al., 2020).
• Generalized Dyck and Fuss-Catalan objects: Order-$\ell$ colored Motzkin paths biject to $k$-Dyck paths with negative floors, Fine paths, and certain $k$-ary trees, with coloring schemes prescribing subclass structure (DeJager et al., 2020).

4. Advanced Applications and Structural Properties

Colored Motzkin paths serve as scaffolds for key enumerative and algebraic identities:

• Enumeration formulas: For $(u,\ell,d)$-colored Motzkin paths of length $n$,

$$M_n^{(u,\ell,d)} = \sum_{r=0}^{\lfloor n/2\rfloor} \frac{1}{r+1}\binom{n}{2r}\binom{2r}{r} u^r \ell^{n-2r} d^r$$

(Woelki, 2013).

• Refined enumeration for two-rowed set-valued tableaux: Closed forms involve factorial and binomial expressions extracted via Lagrange inversion and hypergeometric summation (Krattenthaler, 15 Jan 2025).
• Asymptotics: For colored models, the density of each step type is asymptotic to an explicit algebraic function of the weight parameters (Woelki, 2013).
• Rogers–Ramanujan identities and infinite products: Multicolored dimer models on the segment produce colored Motzkin paths whose generating functions satisfy Fibonacci-type recurrences and, in the thermodynamic limit, exhibit continued product structure and generalized Rogers–Ramanujan identities (Shigechi, 2021).
• Permutations and continued fractions: A variety of permutation statistics (fixed points, cycles, excedances, double-excedances, inversions) correspond to colored Motzkin weights and admit explicit enumeration by continued fractions (Elizalde, 2017).
• Height-bounded walks and amplitude restriction: Enumeration of colored Motzkin paths with maximum height constraint leads to finite continued fractions and correspondences with lattice walks in restricted domains (Courtiel et al., 2020).

5. Structural Refinements and Extensions

Colored Motzkin paths support a range of further extensions, refinements, and subclasses:

• Normalization conditions: For bijections encoding poset intervals, explicit rules (e.g., forbidding $H_2H_1$ at height 0) are mandatory to ensure valid one-to-one mapping (Elizalde et al., 20 Dec 2024).
• Amplitude and boundedness: For applications in walks within simplexes, colored amplitude-bounded Motzkin paths map to lattice walks in triangles or “waffle”-shaped regions, facilitating enumeration via scaffolding algorithms (Courtiel et al., 2020).
• Generalized step sets: Order-$\ell$ colored Motzkin paths allow variable down-step sizes and colored transitions, controlled by vectors; generating functions are Riordan arrays with functional equations parametrized by color schemes (DeJager et al., 2020).
• Automata-theoretic enumeration: Weighted automata approaches, particularly infinite weighted automata, efficiently count generalized Motzkin paths with explicit coloring or weighting objective, especially for $k$-Fibonacci coloring (Castro et al., 2013).
• Combinatorial proofs for classical relations: The 2-to-1 correspondence between large and ordinary colored Motzkin paths yields bijective proofs of classical relations among Schröder numbers, partition numbers, and Catalan families (Chen et al., 2010, Elizalde, 2017).

6. Connections to Statistical Physics and Probability Models

Colored Motzkin paths appear as natural combinatorial models underpinning integrable systems and particle dynamics:

• TASEP steady state weights: The steady-state weights of parallel-update TASEP configurations with fugacity parameters match exactly the total weight of suitable colored Motzkin paths, with step weights reflecting microscopic jump and occupation probabilities (Woelki, 2013).
• Multi-colored dimers and correlation functions: The enumeration of colored Motzkin paths underlies the calculation of emptiness formation probabilities and moments in one-dimensional dimer models; the associated generating functions exhibit profound connections to q-series and partition identities (Shigechi, 2021).
• Phase transition asymptotics: Dominant singularities in colored Motzkin generating functions yield explicit asymptotics for large $n$ (e.g., transfer theorems), with direct physical interpretations in group velocities, fluctuations, and current-density relations in exclusion processes (Woelki, 2013).
• Weight-preserving correspondences: Motzkin–Dyck bijections and expansions enable calculation of moments, transforms, and partition statistics by passing between colored paths and their associated Dyck or Schröder analogs (Shigechi, 2021).

7. Summary and Research Directions

Colored Motzkin paths provide a powerful combinatorial and analytic framework, facilitating explicit enumeration, bijective correspondences, and applications in algebra, geometry, probability, and statistical mechanics. Key recent contributions include explicit generating functions for interval-closed poset sets via bicolored paths (Elizalde et al., 20 Dec 2024), refined bijections to tableaux (Krattenthaler, 15 Jan 2025), continuous parameterization via Riordan arrays (DeJager et al., 2020), rich algebraic identities in dimer models (Shigechi, 2021), and recursive continued fraction representations for permutation statistics (Elizalde, 2017). Open problems include structural normalization for bijections in bounded domain walks (Yeats, 2014), precise amplitude-preserving correspondences, and extensions to multi-parameter weighted analogs in higher dimensions or under further combinatorial constraints.