Motzkin Paths: Properties and Applications

Updated 20 November 2025

Motzkin paths are discrete lattice paths composed of up (U), down (D), and horizontal (H) steps that stay nonnegative, and are counted by the classic Motzkin numbers.
They play a central role in combinatorics with applications in RNA structure modeling, poset enumeration, and bijections with trees and partitions.
Recent research explores their algebraic structure and refined statistics through generating functions, bijections, and generalizations such as colored and higher-order variants.

A Motzkin path is a discrete lattice path in the plane, beginning at $(0,0)$ and ending at $(n,0)$ , that never goes below the horizontal axis and is composed of steps of three kinds: up-step $U=(1,1)$ , down-step $D=(1,-1)$ , and level-step (horizontal) $H=(1,0)$ . Motzkin paths are central objects in enumerative combinatorics, with generalizations and deep connections to lattice path theory, algebra, representation theory, and combinatorial models of physical systems.

1. Classical Motzkin Paths: Definition and Enumerative Properties

A Motzkin path of length $n$ is a sequence of steps from the set $\{U, H, D\}$ , which, starting at $(0,0)$ , traverses the lattice such that the height never becomes negative and terminates at $(n,0)$ . Motzkin paths generalize Dyck paths by allowing horizontal steps in addition to the up and down steps $[2207.07664]$ .

The classical Motzkin numbers $M_n$ count the total number of Motzkin paths of length $n$ , with initial values: $M_0 = 1,\; M_1=1,\; M_2=2,\; M_3=4,\; M_4=9,\; M_5=21,\ldots$ The ordinary generating function is given by: $M(x) = \sum_{n\ge 0} M_n x^n = \frac{1-x-\sqrt{1-2x-3x^2}}{2x^2}$ which satisfies the quadratic equation $M(x) = 1 + x M(x) + x^2 M(x)^2$ $[2207.07664], [1801.04809]$ .

Motzkin numbers and paths appear in numerous combinatorial enumerations, including noncrossing partitions, restricted RNA secondary structures, and the combinatorics of Standard Young Tableaux with three rows $[2104.03774]$ .

2. Generalizations and Variants

Higher-Order and Colored Motzkin Paths

Order- $\ell$ Motzkin paths allow down-steps of depth up to $\ell$ : steps $D_i=(1,-i),\ i\in\{1,\ldots,\ell\}$ , together with up-steps $U=(1,1)$ and level-steps $L=(1,0)$ . These arise in generalized Dyck and Fuss–Catalan families, as well as in relations to $k$ -ary trees and higher genus combinatorics $[2012.14947]$ . Rich coloring and weighting schemes yield generating functions of Riordan-array type, leading to bijections with various families of trees and generalized lattice paths.

Colored and Bicolored Motzkin Paths

In colored Motzkin paths, the level and/or down-steps receive additional "color" labels. (For example, in (3,2)-Motzkin paths, there are three types of level-steps and two types of down-steps.) Such structures are central to bijections with noncrossing linked partitions and variants of the Schröder numbers $[1009.0176], [2412.16368]$ .

A "bicolored Motzkin path" uses two colors for horizontal steps (e.g., "color-1" and "color-2" horizontals), supporting bijections with interval-closed sets in product posets and enabling refined multivariate generating function analysis $[2412.16368]$ . This bicolored setup leads to the functional equation: $C(x,y) = 1 + (x+y) C(x,y) + xy C(x,y)^2$ for the generating function of unrestricted bicolored Motzkin paths.

Generalizations by Step Set

G-Motzkin paths and Motzkin paths with air-pockets further extend the classical definition. For example, G-Motzkin paths permit additional vertical steps $v=(0,-1)$ and are counted via multivariate generating functions and Riordan arrays $[2201.09231]$ . Air-pocket Motzkin paths allow for large "down steps" of arbitrary depth, but forbid consecutive such steps, and their enumeration leverages kernel methods, Lagrange inversion, and bijections with certain Dyck classes $[2301.10449]$ .

3. Connection to Algebraic, Enumerative, and Geometric Structures

Motzkin Paths and Posets

Motzkin paths encode various poset structures. Notably, the Tamari order on Motzkin paths—an extension of the Tamari lattice on Dyck paths—is introduced by defining a partial order via "valley moves," resulting in each connected component of the Motzkin poset being isomorphic to an interval in the classical Tamari lattice $[1801.04809]$ . The number of connected components equals Fibonacci numbers, and intervals are counted via algebraic generating functions.

Motzkin Paths and Vector Spaces

A recent development is the explicit combinatorial map from subspaces of an $n$ -dimensional $q$ -vector space to Motzkin paths of length $n$ $[2407.06559]$ . Every subspace is associated to a Motzkin path via the pattern of left and right pivots of a row-reduced echelon form matrix (assigning up, down, or horizontal steps accordingly). The fiber over a given Motzkin path decomposes into disjoint unions of symmetric Boolean algebras, leading to symmetric chain decompositions and explicit expansions for $q$ -binomial coefficients: $\left[\begin{matrix} n \ k \end{matrix}\right]_q = \sum_{P \in \mathrm{Motz}(n)} (q-1)^{|P|} w(P,q) \left[\begin{matrix} n-2|P| \ k-|P| \end{matrix}\right]_q$ with $w(P,q)$ an explicit product of weights associated to horizontal and down-steps.

Motzkin Paths and Involutions

A fundamental bijection, due to Biane, associates involutions in the symmetric group $S_n$ to labeled Motzkin paths. This is exploited in the rank generating function for the Bruhat order on involutions, with Motzkin path statistics capturing the order-theoretic structure: $R_{I_n}(q) = \sum_{\sigma\in I_n}q^{\mathrm{rk}(\sigma)} = \sum_{\mu\in\mathcal{M}_n}H[\mu;q]$ with rank $H$ expressible in terms of explicit step-level statistics on the Motzkin path $[2106.06021]$ . Specialization to fixed-point-free involutions recovers classical results related to Hermite histories and Dyck paths.

4. Generating Functions, Refined Enumeration, and Asymptotics

Closed-Form Generating Functions

The classical Motzkin numbers have the generating function: $M(x) = \frac{1 - x - \sqrt{1-2x-3x^2}}{2x^2}$ with numerous multivariate and refined analogues in the literature, including those capturing statistics such as the number of horizontal steps at a given level, peak/valley statistics, and colorings $[2201.09231], [2207.07664], [2301.10449]$ .

Many generalizations are encoded by functional equations solved by kernel methods, Lagrange inversion, or even continued fraction expansions in the case of bounded-height, pattern-avoiding, or colored Motzkin families.

Asymptotic Behavior and Scaling Laws

The area-width generating function for Motzkin paths (weighted by area under the path) exhibits universal Airy-type scaling near the tri-critical point. All directed polygon models in the same universality class (Dyck, Motzkin, Schröder) display scaling exponents $1/3,2/3$ and a scaling function given by a quotient of Airy function and its derivative $[1605.09643]$ .

Bivariate and multivariate generating functions enumerate Motzkin paths refined by counts of up, down, and level steps of given types and levels, with precise binomial–Catalan convolutions and Riordan-array representations $[2201.09231]$ . Similar constructions are employed for pattern-avoidance constraints, leading to explicit expressions for the number of paths avoiding specified consecutive subwords $[2310.12497]$ .

Enumerations with forbidden peak heights, valley heights, or run lengths are systematically handled using dynamic programming or symbolic generating functions, often via computer algebra tools $[2010.02389]$ .

5. Bijections and Structural Correspondences

Motzkin paths are in bijection with various classical objects, and this structural flexibility underpins their centrality:

Object/Class	Corresponding Motzkin Path Family	Reference
Ternary trees	S-Motzkin paths, paths with equal U,D,H steps	(Prodinger et al., 2019, Gu et al., 2020)
Ordered pairs of ternary trees	T-Motzkin paths	(Prodinger et al., 2019)
Noncrossing linked partitions	Large (3,2)-Motzkin paths	(Chen et al., 2010)
RNA abstract π-shapes	Motzkin paths via U/D/H symbol interpretation	(Choi, 2019)
Symmetric chain decompositions of vector posets	Motzkin path-indexed Boolean structure	(Farley et al., 9 Jul 2024)
Interval-closed sets in product posets	Bicolored Motzkin paths with step constraints	(Elizalde et al., 20 Dec 2024)
Permutation involutions (Bruhat order)	Motzkin paths with motif-labeled down-steps	(Coopman et al., 2021)

Direct combinatorial bijections mediate between many of these classes (e.g., ternary trees and S-Motzkin paths, Motzkin and Dyck paths via restricted pattern avoidance, etc.), leveraging canonical decompositions (e.g., first return, kernel insertion) or crossing structure in the case of linked partitions.

6. Applications and Combinatorial Models

Motzkin paths provide enumerative answers in a diversity of applied and theoretical contexts:

RNA secondary structure abstraction: π-shapes (non-nested stack abstractions) correspond to Motzkin paths; their enumeration, component count, and asymptotic statistics are all captured via Motzkin generating functions $[1907.07334]$ .
Poset enumeration: Counting interval-closed sets in grid- or rectangle-posets can be achieved through bicolored Motzkin path generating functions, providing a refined and explicit enumeration of poset ideals and interval-closures $[2412.16368]$ .
Quantum exclusion statistics: Weighted Motzkin paths appear in the expansion of cluster coefficients for systems obeying generalized exclusion statistics, with transfer-matrix methods utilizing the combinatorics of periodic Motzkin paths $[2207.07664]$ .
Young tableaux descent statistics: Motzkin paths serve as an equidistributed target for the cyclic descent statistics in three-row standard Young tableaux, with explicit cyclic-descent maps and bijective proofs using Stanley's shuffle theorem $[2104.03774]$ .

Motzkin path enumeration also arises in the analysis of restricted lattice walks, noncrossing graphs, and in the rank statistics of classical combinatorial posets.

7. Summary of Current Research Directions

Recent advances include:

Explicit connections between algebraic objects and Motzkin paths, such as the subspace lattice decomposition $[2407.06559]$ .
Enumeration of advanced generalizations—such as bicolored, higher-order colored, and air-pocket Motzkin paths—supported by powerful kernel, Riordan-array, and Lagrange-inversion methods.
Analysis of refined statistics and pattern-avoidance, with automated, dynamic programming-based enumeration for broad families of constrained Motzkin paths $[2010.02389], [2310.12497]$ .
Bijections with new or classical objects: ongoing efforts expand the taxonomy of objects corresponding to Motzkin path subclasses, further illuminating deep correspondences in enumerative combinatorics and beyond.

Motzkin paths thus remain a central object of research, with ongoing exploration of their combinatorial, algebraic, geometric, and probabilistic facets continuously yielding new connections and applications across mathematics and theoretical computer science.