Catalan Patterns in Combinatorics

• Catalan patterns are constraint phenomena that add forbidden local configurations to traditional Catalan families such as parenthesizations, binary trees, and Dyck paths.
• They are applied to diverse combinatorial objects, including matchings and permutations, producing refined enumerations like Fuss–Catalan, Raney, and Narayana numbers.
• They leverage techniques like generating trees, bijections, and kernel-method functional equations to derive explicit recurrences and closed-form identities.

Catalan patterns are pattern-avoidance and local-constraint phenomena that isolate subclasses of Catalan families—parenthesizations, binary and plane trees, Dyck paths, permutations, Catalan words, and several kinds of matchings—whose enumerations are Catalan, Fuss–Catalan, Raney, Narayana, ballot, Fibonacci, Schröder, or related sequences. In the literature represented here, the term covers both classical avoidance problems and structural restrictions arising from generalized associativity, arithmetic congruence conditions, local forbidden subwords, and forbidden chord configurations. The common mechanism is that a finite or recursive constraint interacts with a standard Catalan decomposition—first return, root degree, insertion, or block factorization—to produce explicit generating functions, bijections, and closed forms (Hein et al., 2015, Alexandersson et al., 2022).

1. Classical Catalan families as the ambient setting

The ordinary Catalan number

$C_n=\frac{1}{n+1}\binom{2n}{n}$

counts the full parenthesizations of $x_0*x_1*\cdots*x_n$, binary trees with $n$ internal nodes, plane trees with $n+1$ nodes, and Dyck paths of length $2n$ (Hein et al., 2015). These models provide the base spaces on which Catalan patterns are imposed.

A link pattern of size $n$ is a perfect matching of $2n$ cyclically ordered points by non-crossing chords. Its cardinality is again $C_n$, with the standard recurrence

$C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$

Ng’s construction of link patterns via strand insertion and Temperley–Lieb generator preimages realizes West’s Catalan tree, with succession rule $(\ell)\to(2),(3),\dots,(\ell+1)$, and transfers refined Catalan statistics to Dyck paths (Ng, 2013).

Catalan words form another standard Catalan family. In one convention they are words $x_0*x_1*\cdots*x_n$0 on nonnegative integers with $x_0*x_1*\cdots*x_n$1 and $x_0*x_1*\cdots*x_n$2; in a shifted convention they are words on positive integers with $x_0*x_1*\cdots*x_n$3 and $x_0*x_1*\cdots*x_n$4. These conventions are equivalent up to adding $x_0*x_1*\cdots*x_n$5 to every letter. Catalan words are in bijection with Dyck paths via the labels or heights of up-steps (Baril et al., 2019, Mansour et al., 2024).

This shared Catalan background is decisive: a “Catalan pattern” is rarely a pattern in isolation. It is a forbidden local configuration placed on one of these canonical Catalan families, and its effect is measured through how much of the original Catalan recursion survives.

2. Modular Catalan numbers and $x_0*x_1*\cdots*x_n$6-associative pattern classes

Hein and Huang introduced the modular Catalan number $x_0*x_1*\cdots*x_n$7 by replacing ordinary associativity with the $x_0*x_1*\cdots*x_n$8-associative law

$x_0*x_1*\cdots*x_n$9

Two parenthesizations are $n$0-equivalent if one can be transformed into the other by repeatedly applying this law, and $n$1 is the number of $n$2-equivalence classes on the $n$3 parenthesizations of $n$4. Equivalently, $n$5 is the number of connected components under $n$6-rotations of the Tamari lattice on binary trees with $n$7 internal nodes (Hein et al., 2015).

The same paper identifies $n$8-minimal representatives in several Catalan families by finite forbidden patterns. In binary trees, $n$9-minimality is avoidance of the left comb $n+1$0. In plane trees, it is the condition that every non-root node has degree $n+1$1. In Dyck paths, this becomes avoidance of the factor $n+1$2, while the dual $n+1$3-maximal condition corresponds to avoiding $n+1$4. Via the Tamari map from $n+1$5-$n+1$6-$n+1$7-avoiding permutations into binary trees, $n+1$8 counts $n+1$9-$2n$0-$2n$1-avoiding permutations avoiding the ascending pattern $2n$2, and $2n$3 refines this further by imposing avoidance of $2n$4 (Hein et al., 2015).

The generating functions are algebraic. With

$2n$5

one has

$2n$6

In particular,

$2n$7

The coefficients admit both a positive-sum formula and an alternating-sum formula; the latter is

$2n$8

The extremal class structure is also explicit: the largest $2n$9-equivalence class on $n$0-node plane trees has size

$n$1

and if $n$2 with $n$3, then exactly $n$4 classes achieve this maximum (Hein et al., 2015).

In this framework, Catalan patterns are not merely forbidden substructures; they are the minimal representatives of generalized associativity classes. That perspective turns pattern avoidance into a quotient theory of Catalan objects.

3. Arithmetic restrictions, permutations, and Fuss–Catalan refinements

A different Catalan-pattern mechanism arises from arithmetic restrictions on permutations. A permutation $n$5 is mod-$n$6-alternating if

$n$7

for every $n$8. For $n$9, this is the classical parity-alternating condition. Writing $2n$0 for the set of such permutations and $2n$1, the classes avoiding $2n$2 or $2n$3 are equinumerous and satisfy

$2n$4

A last-position-of-$2n$5 decomposition yields

$2n$6

and if $2n$7 with $2n$8, then

$2n$9

For $C_n$0, this is exactly the Fuss–Catalan number

$C_n$1

and more generally $C_n$2 coincides with the Raney numbers $C_n$3 (Alexandersson et al., 2022).

The same phenomenon occurs for subexcedant functions. A subexcedant function is a word $C_n$4 with $C_n$5. Under the Mantaci–Rakotondrajao bijection $C_n$6, the mod-$C_n$7 condition on permutations becomes exactly $C_n$8. Restricting further to the Catalan subfamily

$C_n$9

produces the mod-$C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$0-Catalan family $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$1. In the Dyck-path interpretation, $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$2 is an area sequence, and the condition $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$3 forces the number of boxes in each row above the path to be a multiple of $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$4. Consequently, $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$5 equals the number of Dyck paths of semilength $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$6 that never drop below the line $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$7 (Alexandersson et al., 2022).

The two-pattern theory becomes richer in the parity-alternating case. For $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$8, the classification of $C_n=\sum_{k=0}^{n-1}C_k\,C_{n-1-k}.$9 includes powers of $(\ell)\to(2),(3),\dots,(\ell+1)$0, Fibonacci numbers, quadratic binomial formulas, and Raney-type counts. Representative formulas are

$(\ell)\to(2),(3),\dots,(\ell+1)$1

$(\ell)\to(2),(3),\dots,(\ell+1)$2

$(\ell)\to(2),(3),\dots,(\ell+1)$3

and

$(\ell)\to(2),(3),\dots,(\ell+1)$4

For $(\ell)\to(2),(3),\dots,(\ell+1)$5, many two-pattern classes collapse to the identity or a unique trivial permutation (Alexandersson et al., 2022).

Permutation sorting supplies another Catalan-pattern setting. For the two-stack machine with a $(\ell)\to(2),(3),\dots,(\ell+1)$6-avoiding first stack and a classical second stack, exactly four pairs $(\ell)\to(2),(3),\dots,(\ell+1)$7 of length $(\ell)\to(2),(3),\dots,(\ell+1)$8 give Catalan enumeration: $(\ell)\to(2),(3),\dots,(\ell+1)$9 and in each case

$x_0*x_1*\cdots*x_n$00

The single binomial-transform case is $x_0*x_1*\cdots*x_n$01, where

$x_0*x_1*\cdots*x_n$02

and the unique Schröder case is $x_0*x_1*\cdots*x_n$03, with $x_0*x_1*\cdots*x_n$04 (Baril et al., 2020).

Together these results show that Catalan patterns in permutations arise from more than ordinary pattern avoidance. Congruence classes, stack constraints, and Catalan subexcedant encodings all preserve enough recursive structure to generate Fuss–Catalan and Raney refinements.

4. Catalan words: classical patterns, relations, and vincular constraints

Catalan words support an unusually complete finite-pattern theory. For unordered pairs of length-$x_0*x_1*\cdots*x_n$05 patterns, there are $x_0*x_1*\cdots*x_n$06 cases. Baril, Khalil, and Vajnovszki classify these into superfluous, ultimately constant, closed-form, recurrence, and rational-generating-function classes. Representative enumerations include

$x_0*x_1*\cdots*x_n$07

$x_0*x_1*\cdots*x_n$08

and

$x_0*x_1*\cdots*x_n$09

Six pairs yield $x_0*x_1*\cdots*x_n$10, and six others yield $x_0*x_1*\cdots*x_n$11 (Baril et al., 2019).

The descent statistic admits systematic bivariate refinements. For single patterns $x_0*x_1*\cdots*x_n$12 of length at most three, the generating function

$x_0*x_1*\cdots*x_n$13

is explicit in every case. For $x_0*x_1*\cdots*x_n$14,

$x_0*x_1*\cdots*x_n$15

and for $x_0*x_1*\cdots*x_n$16,

$x_0*x_1*\cdots*x_n$17

In the same work, the sequences $x_0*x_1*\cdots*x_n$18 for $x_0*x_1*\cdots*x_n$19 were reported as not yet recorded in OEIS (Baril et al., 2018).

A more general language uses ordered pairs of relations $x_0*x_1*\cdots*x_n$20, with $x_0*x_1*\cdots*x_n$21. The associated bivariate generating functions

$x_0*x_1*\cdots*x_n$22

are given for all $x_0*x_1*\cdots*x_n$23 nontrivial pairs. Examples include

$x_0*x_1*\cdots*x_n$24

for the Fibonacci class $x_0*x_1*\cdots*x_n$25, $x_0*x_1*\cdots*x_n$26, and $x_0*x_1*\cdots*x_n$27, and

$x_0*x_1*\cdots*x_n$28

for $x_0*x_1*\cdots*x_n$29 and $x_0*x_1*\cdots*x_n$30, where the enumeration is $x_0*x_1*\cdots*x_n$31 and the generating function is independent of $x_0*x_1*\cdots*x_n$32. Several repeated formulas are explained by descent-preserving bijections, such as $x_0*x_1*\cdots*x_n$33 and $x_0*x_1*\cdots*x_n$34 (Baril et al., 2023).

Vincular patterns of type $x_0*x_1*\cdots*x_n$35 and $x_0*x_1*\cdots*x_n$36 lead to further Catalan-pattern classes. On Catalan words, one has

$x_0*x_1*\cdots*x_n$37

as well as

$x_0*x_1*\cdots*x_n$38

for the Motzkin numbers,

$x_0*x_1*\cdots*x_n$39

for the Motzkin left-factor numbers, and

$x_0*x_1*\cdots*x_n$40

The difficult cases are handled by kernel-method functional equations, Chebyshev-polynomial expressions, and continued fractions (Mansour et al., 2024).

Flattened Catalan words supply another refined setting. For consecutive patterns of length $x_0*x_1*\cdots*x_n$41 and $x_0*x_1*\cdots*x_n$42, Shattuck derived multivariate generating functions for trios of patterns and proved the distributional equivalences

$x_0*x_1*\cdots*x_n$43

on the flattened class $x_0*x_1*\cdots*x_n$44 (Shattuck, 15 Feb 2025).

Across these variants, Catalan words exhibit an unusually rigid taxonomy: finite pattern sets yield constant, linear, Fibonacci, Pell, Motzkin, or rational-algebraic classes rather than a diffuse spectrum of behaviors.

5. Matchings, link patterns, and Catalan subclasses of Fishburn objects

In noncrossing matchings, Catalan patterns are often literal forbidden chord configurations. For link patterns on $x_0*x_1*\cdots*x_n$45 points, Ng’s inductive algorithm uses strand insertion $x_0*x_1*\cdots*x_n$46 and the preimages $x_0*x_1*\cdots*x_n$47 under a Temperley–Lieb generator. The resulting generating tree is West’s Catalan tree. Under the Dyck-path correspondence, the exposure number of a link pattern corresponds to the last-descent length, and the interaction number corresponds to the number of peaks minus $x_0*x_1*\cdots*x_n$48. Consequently, the number of link patterns of size $x_0*x_1*\cdots*x_n$49 with interaction number $x_0*x_1*\cdots*x_n$50 is the Narayana number

$x_0*x_1*\cdots*x_n$51

while the number with exposure number $x_0*x_1*\cdots*x_n$52 is

$x_0*x_1*\cdots*x_n$53

(Ng, 2013).

Stoimenow matchings form a Fishburn family rather than a Catalan family, but five size-$x_0*x_1*\cdots*x_n$54 forbidden configurations $x_0*x_1*\cdots*x_n$55 define Catalan-counted subclasses. For each $x_0*x_1*\cdots*x_n$56,

$x_0*x_1*\cdots*x_n$57

The class $x_0*x_1*\cdots*x_n$58 is exactly the nonnesting matchings, and the five classes are Wilf-equivalent. Refined statistics remain Catalan: over $x_0*x_1*\cdots*x_n$59, the maximal-crossing statistic satisfies

$x_0*x_1*\cdots*x_n$60

the Narayana polynomial. The trivariate generating function

$x_0*x_1*\cdots*x_n$61

uses the ballot-number series

$x_0*x_1*\cdots*x_n$62

The $x_0*x_1*\cdots*x_n$63-avoiders are also in bijection with $x_0*x_1*\cdots*x_n$64-free posets, ascent sequences avoiding $x_0*x_1*\cdots*x_n$65, and Fishburn permutations avoiding $x_0*x_1*\cdots*x_n$66 (Lv et al., 11 Sep 2025).

Simultaneous avoidance of several Catalan patterns in Stoimenow matchings was later classified completely. For every nonempty subset $x_0*x_1*\cdots*x_n$67, the generating function

$x_0*x_1*\cdots*x_n$68

is explicit. Typical examples are

$x_0*x_1*\cdots*x_n$69

the generating function of the odd-indexed Fibonacci numbers $x_0*x_1*\cdots*x_n$70, and

$x_0*x_1*\cdots*x_n$71

This produces nine OEIS sequences across the simultaneous-avoidance classes (Lv et al., 16 Sep 2025).

These matching models show that Catalan patterns are not confined to Catalan ambient families. They can also carve Catalan slices out of larger classes such as Fishburn objects, with Narayana and ballot refinements persisting under nontrivial bijections.

6. Higher-parameter extensions, mesh patterns, and Catalan identities

A four-parameter generalization of modular Catalan numbers arises from nonassociative binary operations satisfying depth-sensitive congruence conditions. With parameters $x_0*x_1*\cdots*x_n$72, the number $x_0*x_1*\cdots*x_n$73 of equivalence classes of parenthesizations is encoded by

$x_0*x_1*\cdots*x_n$74

and the universal depth recurrence is

$x_0*x_1*\cdots*x_n$75

In the specialization $x_0*x_1*\cdots*x_n$76, one recovers the $x_0*x_1*\cdots*x_n$77-modular Catalan numbers at left-depth $x_0*x_1*\cdots*x_n$78, with

$x_0*x_1*\cdots*x_n$79

These classes admit interpretations in terms of forbidden combs in binary trees, degree constraints in plane trees, and Dyck paths forbidding $x_0*x_1*\cdots*x_n$80 at height $x_0*x_1*\cdots*x_n$81 (Hein et al., 2018).

Mesh patterns connect Catalan patterns to Catalan’s triangle. On the set $x_0*x_1*\cdots*x_n$82 of $x_0*x_1*\cdots*x_n$83-avoiding permutations, the distributions of the mesh patterns $x_0*x_1*\cdots*x_n$84 and $x_0*x_1*\cdots*x_n$85 are

$x_0*x_1*\cdots*x_n$86

while that of $x_0*x_1*\cdots*x_n$87 is

$x_0*x_1*\cdots*x_n$88

where

$x_0*x_1*\cdots*x_n$89

is the Catalan triangle. The same analysis yields the identity

$x_0*x_1*\cdots*x_n$90

and produces a new Catalan-counted family of sequences $x_0*x_1*\cdots*x_n$91 defined by

$x_0*x_1*\cdots*x_n$92

(Kitaev et al., 2012).

Restricted Dyck paths give another source of Catalan identities. For fixed $x_0*x_1*\cdots*x_n$93 and $x_0*x_1*\cdots*x_n$94, let $x_0*x_1*\cdots*x_n$95 be the Dyck paths of semilength $x_0*x_1*\cdots*x_n$96 of height at most $x_0*x_1*\cdots*x_n$97 with no $x_0*x_1*\cdots*x_n$98-fold consecutive valley pattern $x_0*x_1*\cdots*x_n$99 at elevation $n$00. Their generating function is rational,

$n$01

and because $n$02 whenever $n$03, one obtains the Catalan identity

$n$04

for

$n$05

Equivalently,

$n$06

in the same range (Bernini et al., 5 May 2026).

Catalan patterns also appear in new Catalan-counted word models. Stump’s collection $n$07 consists of words $n$08 on nonnegative integers satisfying

$n$09

for all $n$10, together with the condition that if $n$11 is the first occurrence of $n$12, then there are indices $n$13 with $n$14. Via a bubble-sort normalization, the area-sequence encoding of Dyck paths, and Haglund’s zeta map, one obtains a bijection

$n$15

so $n$16 (Stump, 2014).

A plausible implication is that “Catalan patterns” is best viewed not as a single class of forbidden patterns, but as a research program: identify finite local constraints that preserve a Catalan-type recursive skeleton, determine the associated generating functions, and then transport the resulting structure across the standard web of bijections among Catalan families.