Catalan Words: Combinatorial and Algebraic Insights

• Catalan Words are recursively growth-restricted sequences defined by step-wise growth limitations and are counted by the Catalan numbers.
• They offer bijections to Dyck paths and binary trees, enabling efficient enumeration, generation, and ranking in combinatorial frameworks.
• Their structure underpins advanced algebraic constructs like q-shuffle algebras and PBW bases, providing explicit expressions in quantum combinatorics.

A Catalan word is a sequence governed by recursive growth-restrictions encoded by the condition that entries can increase by at most one at each step, typically starting from a specified initial value. Catalan words constitute one of the principal combinatorial models counted by the Catalan numbers, with far-reaching connections to lattice paths (Dyck words), binary trees, pattern avoidance, symmetric function theory, and noncommutative representation theory. Recent decades have produced a detailed classification of their structural properties, explicit generating functions for numerous statistics, and deep applications in algebraic (quantum) combinatorics.

1. Core Definitions and Classes of Catalan Words

A standard Catalan word of length $n$ is a sequence $w = w_1 w_2 \ldots w_n$, usually over $\mathbb{N}$ or $\mathbb{Z}_{>0}$, satisfying:

• $w_1=0$ (or $w_1=1$ in shifted conventions);
• $w_{i+1} \leq w_i + 1$ for $1 \leq i < n$.

The set of such words of length $n$ is denoted $\mathcal{C}_n$. Classical counting yields $$|\mathcal{C}_n| = C_n = \frac{1}{n+1} \binom{2n}{n}$$, the Catalan number.

Variants include:

• Dyck (Catalan) words in binary alphabet: Words of length $2n$ over $\{0,1\}$ where the prefix sums of $h(0)=+1$, $h(1)=-1$ stay non-negative and sum to zero; these encode Dyck paths and parenthesis structures (Kasa, 2010).
• Growth-restricted nonnegative sequences: Words $w$ with $w_1=0$, $0 \leq w_i \leq w_{i-1} + 1$, and sometimes further constraints based on the position of first occurrence of each value (Stump, 2014).
• Statistically decorated variants: E.g., Catalan words tracking numbers of zeros, ones, letters $i$, or more refined subword patterns (Mansour et al., 2014).

These classes are interrelated by explicit bijections, most notably to Dyck words and trees.

2. Enumeration, Generation, and Ranking Algorithms

The canonical enumeration is via the generating function

$$C(x) = \sum_{n=0}^{\infty} C_n x^n = \frac{1 - \sqrt{1-4x}}{2x},$$

satisfying $C(x) = 1 + x C(x)^2$.

Generation: Dyck words (binary Catalan words) of length $2n$ may be generated in lexicographic order by a recursive algorithm maintaining counts of 0's ($n_0$) and 1's ($n_1$), only appending 1 if $n_1 + 1 \leq n_0$ to enforce the non-negativity constraint on prefix sums. This algorithm visits exactly $C_n$ leaves with $O(n C_n)$ total work (Kasa, 2010).

Ranking and Unranking: Using ballot-path numbers $f(i,j)$ (paths from $(0,0)$ to $(i,j)$ staying $j\leq i$), a Dyck word can be ranked or unranked efficiently by encoding the positions of the 1's and counting how many choices at each position yield lexicographically earlier words. The core recurrence for $f(i,j)$ is

$$f(i,0)=1,\quad f(i,j)=0\text{ if }j>i,\quad f(i,j)=f(i-1,j) + f(i,j-1) \quad(1\leq j < i).$$

These techniques enable efficient uniform sampling and indexing (Kasa, 2010).

A notable derived result is a new alternating binomial recurrence for Catalan numbers:

$$C_{n+1} = 1 + \sum_{k=0}^n (-1)^k \binom{n-k}{k+1} C_{n-k},$$

distinct from the classical convolutional recurrences (Kasa, 2010).

3. Structural Bijections and Applications in Algebra

Catalan words have deep bijective correspondences with classical objects:

• Dyck paths: The $i$-th entry $w_i$ encodes the minimum height of the $i$-th up-step of the path; the set $\mathcal{C}_n$ is in bijection with Dyck paths of semilength $n$ (Shattuck, 7 Dec 2025, Baril et al., 8 Apr 2024).
• Binary trees: Assigning $w_1=1$ (or $0$ in alternative conventions) and enforcing growth restrictions, in-order or pre-order traversals of binary trees yield Catalan words; this underlies their enumerative equivalence (Shattuck, 7 Dec 2025).
• Quantum groups and PBW bases: Catalan words, via $q$-shuffle algebra, provide explicit closed-form expressions for PBW bases (Damiani and Beck types) of $U_q^+(\widehat{\mathfrak{sl}_2})$ and its super analogues. The key is that balanced words in $x,y$ with nonnegative weight-prefix sums are the Catalan words indexing basis elements, weighted by explicit $q$-dependent factors (Terwilliger, 2018, Terwilliger, 2021, Zhong et al., 4 Dec 2024).

These algebraic applications are not merely combinatorial: Catalan words provide the backbone for explicit expressions, for example:

$$b(E_{n\delta+\alpha_0}) = q^{-2n}(q - q^{-1})^{2n} x C_n$$

where $C_n$ is a sum of all length-$2n$ Catalan words in $x,y$ weighted by prescribed $q$-factors (Terwilliger, 2018).

4. Statistics, Pattern Avoidance, and Generating Functions

Fine-grained statistics on Catalan words have been extensively analyzed:

• Runs, Valleys, and Peaks: The total number and distribution of runs of (weak) ascents/descents, valleys, peaks, and their symmetric variants are encoded through generating functions, often taking algebraic or rational forms. Narayana-type formulas for e.g. runs of ascents,

$$r(n,k) = \frac{1}{k} \binom{n-1}{k-1} \binom{n}{k-1},$$

are complemented by explicit asymptotics such as $r(n) \sim 4^n/(2\sqrt{\pi n})$ (Baril et al., 8 Apr 2024, Shattuck, 7 Dec 2025).

• Pattern avoidance (classical, consecutive, vincular): For a pattern $p$ (e.g., $010$ or $121$), one tracks the number of Catalan words avoiding $p$, sometimes with respect to descents or related parameters. The bivariate generating function $D_p(x,y) = \sum_{n,k} d_{n,k}^{(p)} x^n y^k$ is typically a rational or algebraic function, and the method of first-return decomposition enables systematic computation (Baril et al., 2018, Mansour et al., 21 May 2024).
• Pairs of patterns: For two patterns $(\pi,\sigma)$, structural and recursive decompositions index cases where enumeration reduces to trivial, recurrence, or generating function forms, and many OEIS sequences acquire new Catalan-word interpretations (Baril et al., 2019).
• Special word classes: Subfamilies such as the set $\mathcal{L}_n$ or $L_n$ (words with additional symmetry or area constraints) are shown to be in bijection with Catalan structures and admit closed formulas for refined statistics (e.g., number of zeros is given by a generalized Narayana formula) (Stump, 2014, Mansour et al., 2014).

5. Multidimensional and Generalized Catalan Words

Three-dimensional Catalan words: Words on a ternary alphabet $\{x,y,z\}$ where any two-letter projection is a Dyck path are "3D Catalan words." The number of such words projecting to a fixed Dyck path is given by explicit factorized formulas in terms of Motzkin numbers and ballot numbers. Enumeration over Dyck paths by this statistic relates to intricate Motzkin and ballot combinatorics, with closed generating functions for fixed values (Archer et al., 2022).

Other generalizations: Non-minimal starting points, “valley-free” or plateaued Catalan words, and words in the Catalan family satisfying additional constraints (e.g., dropout rules or bracketing conditions) further diversify the theory (Stump, 2014).

6. Advanced Algebraic Structures: $q$-Shuffle Algebras and Representation Theory

Catalan words in $q$-shuffle algebras: The positive part $U_q^+$ of $U_q(\widehat{\mathfrak{sl}_2})$ and its super-analogue admit a realization wherein basis vectors are indexed by Catalan words with multiparameter $q$-weights determined by the partial sums of letter weights $x \mapsto +1$, $y \mapsto -1$ (Terwilliger, 2018, Terwilliger, 2021, Zhong et al., 4 Dec 2024). Key theorels include:

• PBW basis generators expressed as (weighted) sums over all Catalan words of a given length, giving commutative or polynomial subalgebras;
• Closed-form exponential and shuffle identities, e.g.,

$$\exp_{*}\left((q-q^{-1}) \sum_{k\geq 1} x C_{k-1} y\, t^k\right) = 1 + \sum_{k\geq 1} C_k t^k,$$

demonstrate the logarithmic relation between certain Catalan-indexed families.

7. Table: Main Classes of Catalan Words and Their Enumeration

Class	Definition/Constraint	Count
Standard	$w_1=0,\ w_{i+1} \leq w_i+1$	$C_n$
Dyck words	Binary, balanced, all prefix sums $\geq 0$	$C_n$
$\mathcal{L}_n$ ($L_n$)	Drop $\leq1$, bracketing ($k$-1 before and after $k$)	$C_{n-1}$
3D Catalan words	Any 2-letter subword = Dyck path	Motzkin/Ballot numbers
PBW (algebraic)	Balanced in $x,y$, nonneg. partial sums	$C_n$ (weighted)

These enumerations are realized both bijectively (via Dyck paths, trees) and algebraically ($q$-deformations, PBW bases, shuffle products).

8. Significance and Research Directions

Catalan words serve as a universal combinatorial code for the Catalan family, with their recursive structure enabling efficient generation, enumeration, and indexing of objects such as Dyck paths, plane trees, binary trees, and even deep algebraic constructs. The interplay between pattern avoidance and classical combinatorial sequences yields new interpretations of both familiar and novel integer sequences. In quantum algebra, Catalan word techniques yield closed PBW bases and transparent commutation relations, streamlining analysis in $q$-shuffle algebraic settings (Terwilliger, 2018, Zhong et al., 4 Dec 2024).

Open directions include:

• Further generalization to multidimensional Catalan words and avoidance of longer or multiple patterns;
• Unification of pattern-avoidance generating functions within recursive-decomposition frameworks;
• Deeper combinatorial understanding of the $q$-shuffle and quantum group connections;
• The extension of Chebyshev-polynomial and functional-equation methods to wider families (Mansour et al., 2014).

Catalan words thus constitute a foundational structure both in enumerative combinatorics and algebraic combinatorics, and their paper continues to uncover new algebraic and combinatorial phenomena.