Catalan Avoidance Class Overview

• Catalan avoidance class is a combinatorial family defined by forbidding specific local patterns, resulting in an enumeration that follows the Catalan numbers.
• The methodology involves analyzing pattern pairs with sorting devices, using recurrences and generating functions to derive canonical counting formulas.
• Canonical bijections map these classes to Dyck paths, binary trees, and noncrossing partitions, highlighting their broad structural and enumerative significance.

A Catalan avoidance class is a combinatorial family arising in the theory of pattern avoidance, where the set of objects under study—such as permutations, words, matchings, or tableaux—avoid specified local patterns or configurations, and the cardinality of the avoidance class is given by the Catalan numbers $C_n = \frac{1}{n+1}\binom{2n}{n}$. The phenomenon is fundamental to enumerative combinatorics, providing a robust unifying theme among apparently disparate structures. The term "Catalan avoidance class" refers specifically to those pattern-avoidance classes whose enumeration, generating functions, or structural decompositions align with the standard Catalan family, either exactly or through canonical bijections.

1. Characterization via Pattern Pairs and Sorting Devices

Catalan avoidance classes are prominently characterized in the context of two-stack sorting machines, where a permutation is sortable if it avoids prescribed forbidden patterns in intermediate stacks. In the setting of a two-stack right-greedy pattern-avoiding machine, the precise pairs of length-3 classical patterns in the first stack yielding Catalan enumeration are:

• (123, 213)
• (132, 312)
• (231, 321)
• (123, 132)

For each such pair, composing the first stack with a classically 21-avoiding stack yields a class of permutations sortable by the machine and counted by $C_n$ (Baril et al., 2020). Notably, these four pairs exhaust all scenarios among pairs of length-3 patterns where the resulting sortable permutations admit a Catalan enumeration.

2. Enumeration, Generating Functions, and Recurrences

For any Catalan avoidance class, the enumeration of $n$-length objects is given by the Catalan sequence: $$C_n = \frac{1}{n+1}\binom{2n}{n} = \frac{(2n)!}{(n+1)!n!}$$ The standard generating function is

$$C(x) = \sum_{n\ge0} C_n x^n = \frac{1 - \sqrt{1 - 4x}}{2x}$$

with the defining quadratic equation

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

and recurrence

$C_0 = 1, \qquad C_{n+1} = \sum_{i=0}^n C_i C_{n-i}$

When the underlying finer combinatorics (e.g., first letter, root subtree sizes) are considered, the classical Catalan triangle $T(n,k)$ appears, refined by

$$T(n,k) = T(n,k-1) + T(n-1,k) \qquad (2 \leq k \leq n)$$

with $C_n = \sum_{k} T(n, k)$. These refined recurrences are key in classes arising from the "mixed" pattern pair (123, 132), which is related to avoidance of forbidden patterns such as $2314, 3214, 4213$ and the barred pattern $[2413$ (Baril et al., 2020).

3. Canonical Bijections and Structural Correspondence

Catalan avoidance classes are not only enumerated by $C_n$ but also admit bijections to canonical Catalan families such as:

• Dyck paths of semilength $n$
• Binary plane trees with $n$ nodes
• 231-avoiding permutations in $S_n$
• Noncrossing set partitions or matchings

For the three complementary involutive pairs, bijection is achieved via an explicit involution $\mathrm{out}_{\sigma,\hat\sigma}$ relating $(\sigma, \hat\sigma)$-sortable permutations to the 231-avoiding class. For the (123, 132) case, a recursive decomposition by first letter yields a direct correspondence to binary trees where root-subtree sizes trace the Catalan triangle (Baril et al., 2020). This underlies the universal mechanism whereby apparently complex pattern-classes can be realized as standard Catalan objects.

4. Principal Results, Theorems, and Algebraic Encodings

Key structural and enumeration results for Catalan avoidance classes include:

• Proposition 2 (Knuth-type): For partial permutations, sortability by a 12-avoiding stack is equivalent to 213-avoidance, and output is the decreasing rearrangement.
• Theorem 2: For any $\sigma \in S_3$, the map $\mathrm{out}_{\sigma,\hat\sigma}$ is a bijection on $S_n$, with $(\sigma, \hat\sigma)$-sortable classes counted by $C_n$ (which also holds for (132, 312) and (231, 321)).
• Theorem 3: The (123, 132)-sortable permutations coincide with avoidance of $\{2314, 3214, 4213, [2413\}$, with first-letter refinement given by the Catalan triangle.
• Corollary 3: All four key pattern-pair sortable classes have cardinality $C_n$ (Baril et al., 2020).

In all Catalan classes, the functional equations and combinatorics reflect the unique solution to quadratic generating function equations and recursive decompositions, both at the total and at the refined level.

5. Broader Context and Enumeration Boundary

The phenomenon of Catalan avoidance classes is sharply demarcated within the broader landscape of pattern avoidance. While minor alterations to the forbidden patterns or composition constraints produce classes enumerated by the Schröder numbers or their binomial transforms (e.g., (132,231) yields large Schröder numbers), exactly four pairs of length-3 patterns retain true Catalan enumeration (Baril et al., 2020). This boundary reflects a deep symmetry: for instance, all three involutive pairs reduce, under composition, to the 231-avoiding class, while "mixed" pairs demand refined "triangle" combinatorics.

The robustness of the Catalan numbers across various sorting contexts, despite local variations in the avoidance operators or stack implementations, illustrates the pervasiveness of the Catalan family in combinatorial enumeration. Conversely, small deformations immediately depart from the Catalan regime.

6. Connections to Other Catalan Structures and Further Directions

Catalan avoidance classes in the sense described above are part of a much larger ecosystem of combinatorial families with Catalan enumeration and structure:

• Principal avoidance classes of Dyck paths, arch systems, and plane forests under the containment ordering (Albert et al., 2014)
• Parabolic Catalan numbers arising from type $A$ Demazure characters and flagged Schur functions (Proctor et al., 2017)
• Catalan subsets of descending plane partitions, alternating sign matrices, and nonnesting matchings, with corresponding generating tree and equinumerous arguments (Keller et al., 2017)
• Catalan congruences on alternating sign matrices, whose equivalence classes correspond to 321-avoiding permutations (as class minima) and covexillary permutations (as maxima), with direct implications for bases of the Temperley–Lieb algebra (Hivert et al., 10 Nov 2025)

The presence of canonical bijections across these domains further underscores the structural cohesion and universality of Catalan pattern-avoidance classes.

7. Significance in Enumerative and Algorithmic Combinatorics

Catalan avoidance classes encapsulate the relationship between local combinatorial constraints (pattern avoidance) and global structure (enumeration by Catalan numbers), serving as a testbed for bijective, algebraic, and functional-analytic techniques. The sharp boundaries observed—where only special avoidance types preserve Catalan enumeration—highlight the delicate balance between local and global combinatorial phenomena and inform algorithmic questions in sorting and structure generation (Baril et al., 2020). The mapping of these classes into other algebraic and lattice-theoretic frameworks (e.g., congruence classes in distributive lattices, Hecke monoids) further broadens their impact beyond purely enumerative settings.