Paperfolding and Catalan numbers

Two New Identities Involving the Catalan Numbers: A classical approach

Two new identities about Catalan numbers are treated with Zeilberger's algorithm and Watson's hypergeometric series evaluation.

$s$-Catalan numbers and Littlewood-Richardson polynomials

In this note, we study two generalizations of the Catalan numbers, namely the $s$-Catalan numbers and the spin $s$-Catalan numbers. These numbers first appeared in relation to quantum physics problems about spin multiplicities. We give a combinatorial description for these numbers in terms of Littlewood-Richardson coefficients, and explain some of the properties they exhibit in terms of Littlewood-Richardson polynomials.

卡特兰数的移位汉克尔行列式及相关结果 II：后向移位

Shifted Hankel determinants of Catalan numbers and related results II: Backward shifts

该研究通过前置零定义序列后向移位的汉克尔行列式，得到卡特兰数及相关数与多项式序列的相关结果，并对卡特兰数卷积幂序列提出猜想。

Yet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangles

We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, al…

Some identities and formulas involving generalized Catalan numbers

A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.

On the (n,k)-th Catalan numbers

In this paper, we generalize the Catalan number to the $(n,k)$-th Catalan numbers and find a combinatorial description that the $(n,k)$-th Catalan numbers is equal to the number of partitions of $n(k-1)+2$ polygon by $(k+1)$-gon where all vertices of all $(k+1)$-gons lie on the vertices of $n(k-1)+2$ polygon.

Eulerian-Catalan Numbers

We show that the Eulerian-Catalan numbers enumerate Dyck permutations. We provide two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analogue of the Chung-Feller theorem.

Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers

We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the known results for linear combinations of pairs of Catalan numbers. Many interesting number triangles emerge, some of which we analyze using the language of Riordan arrays.

Catalan-like numbers and Hausdorff moment sequences

In this paper we show that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers are Hausdorff moment sequences in a unified approach. In particular we answer a conjecture of Liang at al. which such numbers have unique representing measures. The smallest interval including the support of representing measure is explicitly found. Subsequences of Catalan-like numbers are also considered. We provide a necessary and sufficient condition for a pattern of subsequences that if sequences are the Stielt…

q-Catalan numbers and q-Narayana polynomials

In this note we show that various natural q-analogues of the Catalan numbers can be obtained in a uniform way. Furthermore we compute their Hankel determinants.

卡特兰数序列与广义作用图

Catalan number sequences and generalized action graphs

该研究探讨作用图与多类卡特兰相关序列的关联，证明作用图无法推广至卡特兰三角、$(a,b)$-卡特兰数及内三角，同时提出了为超级卡特兰数构造作用图的猜想方法。

关于与类Catalan数相关的Hankel行列式的若干注记

Some remarks about Hankel determinants which are related to Catalan-like numbers

该研究针对后移类Catalan数及相关序列，得出了其Hankel行列式的相关结果，属于组合数学领域的注记类研究。

A note on super Catalan numbers

We show that the super Catalan numbers are special values of the Krawtchouk polynomials by deriving an expression for the super Catalan numbers in terms of a signed set.

Catalan-like numbers and Stieltjes moment sequences

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schröder numbers, are Stieltjes moment sequences in a unified approach.

The Super Catalan Numbers $S(m, m + s)$ for $s \leq 4$

We give a combinatorial interpretation using lattice paths for the super Catalan number $S(m, m+s)$ for $s \leq 3$ and a separate interpretation for $s = 4$.

Combinatoric results for graphical enumeration and the higher Catalan numbers

We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps.

利用Féaux公式的Catalan数积分表示

An integral representation of Catalan numbers using the Féaux formula

本研究利用Féaux的Gamma函数对数积分表示，推导出Catalan数的一种新积分表达式，该式可为中心二项式系数、Catalan数相关新恒等式的推导提供基础。

三Catalan数的组合性质及若干正性

Combinatorics of three-Catalan numbers and some positivities

本文提出三Catalan三角，利用三Catalan数的递推与组合解释证明其对数凸性，并证明三角各行具有对数凹性。

Super-Catalan Numbers of the Third and Fourth Kind

The Super-Catalan numbers are a generalization of the Catalan numbers defined as $T(m,n) = \frac{(2m)!(2n)!}{2m!n!(m+n)!}$. It is an open problem to find a combinatorial interpretation for $T(m,n)$. We resolve this for $m=3,4$ using a common form; no such solution exists for $m=5$.

The $q$-golden ratio, Catalan numbers, and an identity of Sauermann--Wigderson

In this note, we present some basic properties of $q$-Fibonacci numbers and their relationship to the $q$-golden ratio and Catalan numbers. We then use this relationship to give a short proof of a combinatorial identity.

A note on log-convexity of q-Catalan numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.

Catalan Numbers and Jacobi Polynomials

We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.

Modular Catalan Numbers

The Catalan number $C_n$ enumerates parenthesizations of $x_0*\dotsb*x_n$ where $*$ is a binary operation. We introduce the modular Catalan number $C_{k,n}$ to count equivalence classes of parenthesizations of $x_0*\dotsb*x_n$ when $*$ satisfies a $k$-associative law generalizing the usual associativity. This leads to a study of restricted families of Catalan objects enumerated by $C_{k,n}$ with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for $C_{k,n}$ with two different proofs. For each $n\ge0$ we compute the largest size of $…

Shifted Hankel determinants of Catalan numbers and related results

In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation exists for more general sequences.

Positroid Catalan numbers

Given a permutation $f$, we study the positroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with the generalized $q,t$-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky hom…

A weighted interpretation for the super Catalan numbers

The super Catalan numbers $T(m,n)=(2m)!(2n)!/2m!n!(m+n)!$ are integers which generalize the Catalan numbers. With the exception of a few values of $m$, no combinatorial interpretation in known for $T(m,n)$. We give a weighted interpretation for $T(m,n)$ and develop a technique that converts this weighted interpretation into a conventional combinatorial interpretation in the case $m=2$.

The Limiting Distribution of the Coefficients of the $q$-Catalan Numbers

We show that the limiting distributions of the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $n$, we conjecture that for sufficiently large $n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail.

Catalan numbers and relations

We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to most of the combinatorial interpretations of Catalan numbers. We show, in particular, that the second component R uniquely determines the pair, and we give a characterization of R in terms of forbidden configurations. We also propose some generalizations of Catalan pairs arising from some slight modifications of (some of the) axioms.

Balanced Properties of the q-Derangement Numbers and the q-Catalan Numbers

Based on Bona's condition for the balanced property of the number of cycles of permutations, we give a general criterion for the balanced property in terms of the generating function of a statistic. We show that the q-derangement numbers and the q-Catalan numbers satisfy the balanced property.

Generalized Catalan numbers

A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial cases. This $ q $-deformed version will likely categorify series arise from identities of the Roger-Ramanujan type.

Some observations about determinants which are connected with Catalan numbers and related topics

In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and connections with Hankel determinants. Finally we state some conjectures.

A derivation of the Catalan numbers from a bijection between permutations and labeled trees

We present an algorithmic mapping from permutations of length dn to labeled n-node d-ary trees and back again. Given such a bijection, one can interpret each of the factorials in the formula for the Catalan numbers as a count of permutations.

卡塔兰数的解析推广及其积分表示

An analytic generalization of the Catalan numbers and its integral representation

本文对组合数论中的卡塔兰数进行解析推广，并利用柯西积分公式给出其积分表示，同时指出后续研究方向。

Partitions, pairs of trees and Catalan numbers

In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a single square partition or with several partitions with only one part. The cardinalities of those families of partitions are the Catalan and ballot numbers, respectively. In this paper we present a closed description for those families. We also present bijections between those sets of partitions and sets of trees and forests enumerated by the Catalan an ballot nu…

Variations of the Catalan numbers from some nonassociative binary operations

We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.

Three-dimensional Catalan numbers and product-coproduct prographs

We present the new combinatorial class of product-coproduct prographs which are planar assemblies of two types of operators: products having two inputs and a single output and coproducts having a single input and two outputs. We show that such graphs are enumerated by the $3$-dimensional Catalan numbers. We present some combinatorial bijections positioning product-coproduct prographs as key objects to probe families of objects enumerated by the $3$-dimensional Catalan numbers.

Random graphs induced by Catalan pairs

We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce various properties of these random graphs. In particular, we asymptotically determine the expected number of edges and isolated vertices, and more generally we determine the expected number of (induced) subgraphs isomorphic to a given connected graph.

On Powers of the Catalan Number Sequence

The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the first time that it is an infinitely divisible Stieltjes moment sequence in the sense of S.-G. Tyan. Besides, any positive real power of the sequence is still a Stieltjes determinate sequence. Some more cases including (a) the central binomial coefficient sequence (related to the Catalan sequence), (b) a double factorial number sequence and (c) the generalized…

Determinants of (generalised) Catalan numbers

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This approach leads also naturally to extensions and generalisations.

A matrix with sums of Catalan numbers -- LU-decomposition and determinant

Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help. Specializing a parameter, the determinant turns out to be a Fibonacci number with odd index, confirming earlier results, obtained back then by combinatorial methods.

关于卡特兰数卷积幂的汉克尔行列式的若干实验观察

Some experimental observations about Hankel determinants of convolution powers of Catalan numbers

本研究通过计算机实验提出了若干关于卡特兰数卷积幂汉克尔行列式的猜想，虽多数暂无证明，但仍予以公布以期学界后续证明。

基于半整数二项式系数的调和卡特兰数级数

Harmonic Catalan Number Series via Half-Integer Binomial Coefficients

该研究通过半整数参数的交替二项式恒等式，结合广义二项式系数的显式表示与相关恒等式，推导出含卡特兰数、调和数的闭式求和公式，为转化恒等式提供了系统机制。

Some identities for the Catalan and Fine numbers

We establish combinatorial interpretations of several identities for the Catalan and Fine numbers and, along the way, we present some new bijections of independent interest. Briefly, we show that C_{n} = 1/(n+1) Sum_{k} (n+1)choose(2k+1) (n+k)choose(k) counts ordered trees on n edges by number of interior vertices adjacent to a leaf, and C_{n} = 2/(n+1) Sum_{k} (n+1)choose(k+2) (n-2)choose(k) counts Dyck n-paths by number of long interior inclines. We also give an analogue for the Fine numbers of Touchard's Catalan number identity.

Action graphs, planar rooted forests, and self-convolutions of the Catalan numbers

We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.

Harmonic numbers, Catalan's triangle and mesh patterns

The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we link avoidance of one of the patterns to the harmonic numbers, while for three other patterns we show their distributions on 132-avoiding permutations are given by the Catalan triangle. Also, we show that two specific mesh patterns are Wilf-equivalent. As a byproduct of our studies, we define a new set of sequences counted by the Catalan numbers and provide a r…

Action graphs and Catalan numbers

We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step $k$ is equal to the $k$th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step $k$ and the set of planar rooted trees with $k$ edges.

Rational parking functions and Catalan numbers

The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we prop…

(k,m)-Catalan Numbers and Hook Length Polynomials for Plane Trees

Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is C_{k,m}(n)=\frac{1}{mn+1}{(mn+1)k \choose n}, which we call the (k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)-ary trees, which are counted by the (k,m)-Catalan numbers.

The Super Patalan Numbers

We introduce the super Patalan numbers, a generalization of the super Catalan numbers in the sense of Gessel, and prove a number of properties analagous to those of the super Catalan numbers. The super Patalan numbers generalize the super Catalan numbers similarly to how the Patalan numbers generalize the Catalan numbers.

Modular Fuss-Catalan numbers

The modular Catalan numbers $C_{k,n}$, introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of $x_0 * x_1 * \dots *x_n$ where $*$ is a binary $k$-associative operation and $k$ is a positive integer. The classical notion of associativity is just 1-associativity, in which case $C_{1,n} = 1$ and the size of the unique class is given by the Catalan number $C_n$. In this paper we introduce modular Fuss-Catalan numbers $C_{k,n}^{m}$ which count equivalence classes of parenthesizations of $x_0 * x_1 * \dots *x_n$ where $*$ is an $m$-ary $k$-associative operation for $m …

On super Catalan polynomials

We present a $q$-analog of the super Catalan number $(2m)!(2n)!/2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(λ)$, due to Fürlinger and Hofbauer, for $λ=0$ and $λ=1$. We give a combinatorial interpretation for this analog when $m=2$.

Knight's paths towards Catalan numbers

We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the $x$-axis are enumerated by the generalized Catalan numbers, and we give a constructive bijection with peakless Motzkin paths of a given length. After enumerating partial knight's paths of a given length, we prove that zigzag knight's paths of a given length ending on the $x$-axis are counted by the Catalan numbers. Finally, we give a constructive bijection with Dyck paths of a given length.

Identities involving Narayana polynomials and Catalan numbers

We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.

A determinant representation for generalized ballot and Fuss-Catalan numbers

In this note we introduce a determinant and then give its evaluating formula. The determinant turns out to be a generalization of the well-known ballot and Fuss-Catalan numbers, which is believed to be new. The evaluating formula is proved by showing that the determinant coincides with the number of lattice paths with (1,0), (0,1)-steps in the plane that stay below a boundary line of rational slope.