--- title: Catalan Patterns in Combinatorics url: https://www.emergentmind.com/topics/catalan-patterns type: topic --- # Catalan Patterns in Combinatorics 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 [1508.01688][2201.08168]. ## 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 \(2n\) [1508.01688]. 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 [1305.4877]. Catalan words form another standard Catalan family. In one convention they are words \(w=w_1\cdots w_n\) on nonnegative integers with \(w_1=0\) and \(0\le w_i\le w_{i-1}+1\); in a shifted convention they are words on positive integers with \(w_1=1\) and \(w_{i+1}\le w_i+1\). These conventions are equivalent up to adding \(1\) to every letter. Catalan words are in bijection with Dyck paths via the labels or heights of up-steps [1912.10447][2405.12435]. 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 \(k\)-associative pattern classes Hein and Huang introduced the modular Catalan number \(C_{k,n}\) by replacing ordinary associativity with the \(k\)-associative law \[ (x_0*x_1*\cdots*x_k)*x_{k+1}=x_0*(x_1*x_2*\cdots*x_{k+1}). \] Two parenthesizations are \(k\)-equivalent if one can be transformed into the other by repeatedly applying this law, and \(C_{k,n}\) is the number of \(k\)-equivalence classes on the \(C_n\) parenthesizations of \(x_0*\cdots*x_n\). Equivalently, \(C_{k,n}\) is the number of connected components under \(k\)-rotations of the Tamari lattice on binary trees with \(n\) internal nodes [1508.01688]. The same paper identifies \(k\)-minimal representatives in several Catalan families by finite forbidden patterns. In binary trees, \(k\)-minimality is avoidance of the left comb \(\mathrm{comb}_k^1\). In plane trees, it is the condition that every non-root node has degree \(1) \] produces the mod-\(k\)-Catalan family \(MP_C(n,k)\). In the Dyck-path interpretation, \((f_i-1)\) is an area sequence, and the condition \(f_i\equiv i\pmod{k}\) forces the number of boxes in each row above the path to be a multiple of \(k\). Consequently, \(|MP_C(n,k)|\) equals the number of Dyck paths of semilength \(n\) that never drop below the line \(y=kx\) [2201.08168]. The two-pattern theory becomes richer in the parity-alternating case. For \(k=2\), the classification of \(MP_{\sigma,\tau}(n,2)\) includes powers of \(2\), Fibonacci numbers, quadratic binomial formulas, and Raney-type counts. Representative formulas are \[ MP_{132,123}(2m,2)=2^{m-1},\qquad MP_{132,123}(2m+1,2)=1, \] \[ MP_{213,132}(2m,2)=2^{m-1},\qquad MP_{213,132}(2m+1,2)=F_{2m+1}, \] \[ MP_{321,132}(2m,2)=MP_{321,132}(2m+1,2)=\binom{m}{2}+1, \] and \[ MP_{312,231}(n,2)=F_n. \] For \(k\ge3\), many two-pattern classes collapse to the identity or a unique trivial permutation [2201.08168]. Permutation sorting supplies another Catalan-pattern setting. For the two-stack machine with a \(\sigma\)-avoiding first stack and a classical second stack, exactly four pairs \((\sigma,\tau)\) of length \(3\) give Catalan enumeration: \[ (\sigma,\tau)=(123,213),\ (132,312),\ (231,321),\ (123,132), \] and in each case \[ |Sort_n(\sigma,\tau)|=C_n. \] The single binomial-transform case is \((123,312)\), where \[ |Sort_{n+1}(123,312)|=\sum_{k=0}^n \binom{n}{k}C_k, \] and the unique Schröder case is \((132,231)\), with \(|Sort_n(132,231)|=S_n\) [2004.01812]. 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-\(3\) patterns, there are \(\binom{10}{2}=45\) cases. Baril, Khalil, and Vajnovszki classify these into superfluous, ultimately constant, closed-form, recurrence, and rational-generating-function classes. Representative enumerations include \[ c_n(000,101)=2^{n-1}, \qquad c_n(001,210)=\binom{n}{3}+n, \] \[ c_n(000,001)=F_n, \qquad c_n(100,201)=F_{2n-2}, \qquad c_n(100,101)\ \text{is the }n\text{th Pell number}, \] and \[ c_n(011,100)=c_n(011,120)=2(n-1). \] Six pairs yield \(c_n=n\), and six others yield \(c_n=\binom{n}{2}+1\) [1912.10447]. The descent statistic admits systematic bivariate refinements. For single patterns \(p\) of length at most three, the generating function \[ G_p(x,y)=\sum_{n\ge0}\sum_{k\ge0} a_{n,k}(p)\,x^n y^k \] is explicit in every case. For \(p\in\{012,001\}\), \[ G_p(x,y)=\frac{1-x+x^2-x^2y}{1-2x+x^2-x^2y}, \] and for \(p=021\), \[ G_{021}(x,y)=\frac{1 - 4x + 6x^2 - x^2y - 4x^3 + 3x^3y + x^4 - x^4y} {(1-x)(1-2x)(1-2x+x^2-x^2y)}. \] In the same work, the sequences \(|\mathcal C_n(p)|\) for \(p\in\{000,110,210\}\) were reported as not yet recorded in OEIS [1803.06706]. A more general language uses ordered pairs of relations \((X,Y)\), with \(X,Y\in\{<,>,\le,\ge,=,\neq\}\). The associated bivariate generating functions \[ G_{X,Y}(x,y)=\sum_{w\text{ avoids }(X,Y)} x^{|w|}y^{des(w)} \] are given for all \(16\) nontrivial pairs. Examples include \[ G_{7}(x,y)=\frac{1+x^2-x^2y}{1-x-x^2y} \] for the Fibonacci class \((\ge,\le)\), \((\le,<)\), and \((<,\le)\), and \[ G_{12}(x,y)=\frac{1-x+x^2}{(1-x)^2} \] for \((<,\ge)\) and \((\neq,\ge)\), where the enumeration is \(n\) and the generating function is independent of \(y\). Several repeated formulas are explained by descent-preserving bijections, such as \((\ge,\ge)\leftrightarrow(<,<)\) and \((\ge,>)\leftrightarrow(>,<)\) [2302.12741]. Vincular patterns of type \((1,2)\) and \((2,1)\) lead to further Catalan-pattern classes. On Catalan words, one has \[ c_n(1\hyphen12)=c_n(1\hyphen21)=c_n(1\hyphen23)=2^{n-1}, \qquad c_n(2\hyphen12)=F_{2n-1}, \qquad c_n(3\hyphen12)=C_n, \] as well as \[ c_n(1\hyphen22)=M_n \] for the Motzkin numbers, \[ c_n(2\hyphen31)=L_n \] for the Motzkin left-factor numbers, and \[ c_n(12\hyphen2)=1+\binom{n}{2}. \] The difficult cases are handled by kernel-method functional equations, Chebyshev-polynomial expressions, and continued fractions [2405.12435]. Flattened Catalan words supply another refined setting. For consecutive patterns of length \(2\) and \(3\), Shattuck derived multivariate generating functions for trios of patterns and proved the distributional equivalences \[ 112\approx122,\qquad 211\approx221\approx231 \] on the flattened class \(\mathcal F_n\) [2502.10661]. 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 \(2n\) points, Ng’s inductive algorithm uses strand insertion \(\mathrm{Ins}_i\) and the preimages \(M_0,M_1,\dots,M_k\) 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 \(1\). Consequently, the number of link patterns of size \(n\) with interaction number \(\ell\) is the Narayana number \[ N(n,\ell+1)=\frac{1}{n}\binom{n}{\ell+1}\binom{n}{\ell}, \] while the number with exposure number \(k\) is \[ \binom{2n-k}{n}\cdot \frac{k}{2n-k} \] [1305.4877]. Stoimenow matchings form a Fishburn family rather than a Catalan family, but five size-\(4\) forbidden configurations \(P_1,\dots,P_5\) define Catalan-counted subclasses. For each \(i=1,\dots,5\), \[ |\mathcal M_n(P_i)|=C_n=\frac{1}{n+1}\binom{2n}{n}. \] The class \(\mathcal M_n(P_1)\) is exactly the nonnesting matchings, and the five classes are Wilf-equivalent. Refined statistics remain Catalan: over \(\mathcal M_n(P_1)\), the maximal-crossing statistic satisfies \[ \sum_{M\in\mathcal M_n(P_1)} x^{\mcr(M)} = N_n(x) = \frac1n\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1}x^k, \] the Narayana polynomial. The trivariate generating function \[ \sum_{M\in\mathcal M(P_1)} y^{\fcr(M)}z^{\bl(M)}t^{|M|} = 1+yzt\,C(y,t)\,C(z,t), \] uses the ballot-number series \[ C(x,t)=\sum_{n\ge0}\sum_{k\ge0} B(n,k)x^k t^n, \qquad B(n,k)=\frac{k}{2n-k}\binom{2n-k}{n}. \] The \(P_2\)-avoiders are also in bijection with \((2+2,N)\)-free posets, ascent sequences avoiding \(101\), and Fishburn permutations avoiding \(3142\) [2509.09115]. Simultaneous avoidance of several Catalan patterns in Stoimenow matchings was later classified completely. For every nonempty subset \(S\subseteq\{P_1,\dots,P_5\}\), the generating function \[ A_S(x)=\sum_{n\ge0} |\mathcal M_n(S)|\,x^n \] is explicit. Typical examples are \[ A_{\{P_2,P_4\}}(x)=A_{\{P_2,P_5\}}(x)=A_{\{P_3,P_4\}}(x)=A_{\{P_3,P_5\}}(x)=A_{\{P_4,P_5\}}(x)=\frac{1-2x}{1-3x+x^2}, \] the generating function of the odd-indexed Fibonacci numbers \(F_{2n-1}\), and \[ A_{\{P_1,\dots,P_5\}}(x)=\frac{1-2x+2x^2+x^3}{(1-x)^3}. \] This produces nine OEIS sequences across the simultaneous-avoidance classes [2509.12726]. 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 \((d,e,k,\ell)\), the number \(C^{d,e}_{k,\ell,n}\) of equivalence classes of parenthesizations is encoded by \[ G_{d,e;k,\ell}(x)=\sum_{n\ge0} C^{d,e}_{k,\ell,n}\,x^{n+1}, \] and the universal depth recurrence is \[ G_{d,e;k,\ell}(x)=x+G_{d-1,e;k,\ell}(x)\,G_{d,e-1;k,\ell}(x). \] In the specialization \(e=\ell=1\), one recovers the \(k\)-modular Catalan numbers at left-depth \(d\), with \[ C^{d+1}_k(x)=\frac{x}{1-C_k^d(x)}, \qquad C_k(x)=C_k^1(x)=\frac{x(1-x^{k-1})}{1-2x+x^k}. \] These classes admit interpretations in terms of forbidden combs in binary trees, degree constraints in plane trees, and Dyck paths forbidding \(DU^k\) at height \(\ge d\) [1807.04623]. Mesh patterns connect Catalan patterns to Catalan’s triangle. On the set \(S_n(132)\) of \(132\)-avoiding permutations, the distributions of the mesh patterns \(s_1\) and \(s_2\) are \[ C(n-1,k), \] while that of \(s_3\) is \[ C(n-1,n-1-k), \] where \[ C(n,k)=\frac{(n+k)!\,(n-k+1)}{k!\,(n+1)!} \] is the Catalan triangle. The same analysis yields the identity \[ C(n,k)=\sum_{i=0}^{k} C_i\,C(n-i-1,k-i), \] and produces a new Catalan-counted family of sequences \((a_1,\dots,a_n)\) defined by \[ a_1=0,\qquad 0\le a_i\le i-1-\sum_{j=1}^{i-1} a_j \quad (i=2,\dots,n) \] [1209.6423]. Restricted Dyck paths give another source of Catalan identities. For fixed \(h\ge3\) and \(k\ge2\), let \(\mathcal D_n^{(h,k)}\) be the Dyck paths of semilength \(n\) of height at most \(h\) with no \((k-1)\)-fold consecutive valley pattern \((DU)^{k-1}\) at elevation \(h-1\). Their generating function is rational, \[ D^{(h,k)}(x)=\frac{P^{(h,k)}(x)}{S^{(h,k)}(x)}, \] and because \(D_n^{(h,k)}=C_n\) whenever \(n0\) is the first occurrence of \(k\), then there are indices \(i_1