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Catalan numbers, q-Catalan numbers and hypergeometric series

George E Andrews
1987 Journal of combinatorial theory. Series A
Catalan numbers are examined in the context of hypergeometric series. We are thus able to produce new and simple q-analogs related to the theory of partitions.  ...  INTRODUCTION III a recent paper, Fiirlinger and Hofbauer  present a survey and unification of the q-analogs of Catalan numbers.  ...  In the introduction they point out that the Catalan numbers C, most often arise from combinatorial problems as solutions of the recurrence n-1 c,= c CkCn-k, c, = 1.  ...

Parametric Catalan numbers and Catalan triangles

Tian-Xiao He
2013 Linear Algebra and its Applications
Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder  ...  Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays.  ...  e)−Catalan numbers.  ...

Catalan numbers revisited

Daniel Rubenstein
1994 Journal of combinatorial theory. Series A
THE CATALAN NUMBERS AND PAIRED PARENTHESES /__\ We define the nth Catalan number to be C n = ~z~l/(n + 1).  ...  the number of paired open parentheses equals the number of paired closed parentheses, the number of unpaired open parentheses equals the number of unpaired closed parentheses.  ...

Generalized Catalan Numbers [chapter]

2011 Fibonacci and Catalan Numbers
In this paper we present an algebraic interpretation for generalized Catalan numbers. We describe them as dimensions of certain subspaces of multilinear polynomials.  ...  Since C(n, n) = C n , they obviously generalize Catalan numbers, and they are referred as generalized Catalan numbers.  ...  Truncated stack-sortable permutations and sequences The sequence (C n ) n∈N of Catalan numbers where the n-th Catalan number C n is defined as 1 n + 1 2n n occurs in many different situations.  ...

Eulerian-Catalan Numbers [article]

Hoda Bidkhori, Seth Sullivant
2011 arXiv   pre-print
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.  ...  (ii) Both the Catalan numbers and Eulerian numbers have q (and q, t) analogues. Do these extend to the Eulerian-Catalan numbers?  ...

Noncommutative Catalan numbers [article]

2019 arXiv   pre-print
The goal of this paper is to introduce and study noncommutative Catalan numbers \$C_n\$ which belong to the free Laurent polynomial algebra in \$n\$ generators.  ...  Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman \$(q,t)\$-versions, another -- to solving noncommutative quadratic equations  ...  The following result shows that our (truncated) noncommutative Catalan numbers are noncommutative deformations of (q, 1)-Catalan numbers.  ...

q-Catalan numbers

J Fürlinger, J Hofbauer
1985 Journal of combinatorial theory. Series A
most obvious q-analog of the Catalan numbers.  ...  They will be the starting point of our study of q-Catalan numbers.  ...

Eulerian-Catalan Numbers

Hoda Bidkhori, Seth Sullivant
2011 Electronic Journal of Combinatorics
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.  ...  Both the Catalan numbers and Eulerian numbers have q (and q, t) analogues. Do these extend to the Eulerian-Catalan numbers? 3.  ...

Generalized Catalan numbers [article]

Keke Zhang
2020 arXiv   pre-print
A finitization of the Catalan numbers \$ C_n \$ can be defined as Euler characteristics of an algebraic structure.  ...  For example, we have the following: Let C n denote the nth Catalan number.  ...  When we set N → ∞, we have the q-dimension to be the q-Catalan numbers.  ...

Positroid Catalan numbers [article]

Pavel Galashin, Thomas Lam
2021 arXiv   pre-print
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.  ...  Recurrence for positroid Catalan numbers.  ...

Modular Catalan Numbers [article]

Nickolas Hein, Jia Huang
2016 arXiv   pre-print
The Catalan number \$C_n\$ enumerates parenthesizations of \$x_0*\dotsb*x_n\$ where \$*\$ is a binary operation.  ...  For each \$n\ge0\$ we compute the largest size of \$k\$-associative equivalence classes and show that the number of classes with this size is a Catalan number.  ...  This Catalan number gives an upper bound for the number of ways to interpret the meaning of x 0 * · · · * x n .  ...

Catalan numbers revisited

Gloria Olive
1985 Journal of Mathematical Analysis and Applications
Since the Catalan number al; has many interesting combinatorial interpretations [as indicated in Section 11, it seems reasonable to expect that other !Bt will have some nice interpretations.  ...  We observe that the number of r -V, P is (;I:). The following theorem reveals that the number of r -Vh PT is a binomial CF(1). THEOREM 8A.l. &?\$h+r= # of r -V,,PT. ProoJ: We first note that 1 <h d r.  ...

Remarks on Catalan and super-Catalan numbers

Anna Dorota Krystek, Łukasz Jan Wojakowski
2011 Banach Center Publications
In this article we discuss the Catalan and super-Catalan (or Schröder) numbers. We start with some combinatorial interpretations of those numbers.  ...  We study two probability measures in the context of free probability, one whose moments are super-Catalan, and another, whose even moments are super-Catalan and odd moments are zero.  ...  Interpretations of Catalan and super-Catalan numbers 1.  ...

Chromatic statistics for Catalan and Fuß-Catalan numbers [article]

Roland Bacher, Christian Krattenthaler
2011 arXiv   pre-print
We refine Catalan numbers and Fu{\ss}-Catalan numbers by introducing colour statistics for triangulations of polygons and \$d\$-dimensional generalisations there-of which we call Fu{\ss}-Catalan complexes  ...  Our refinements consist in showing that the number of triangulations, respectively Fu{\ss}-Catalan complexes, with a given colour distribution of its vertices is given by closed product formulae.  ...  These numbers are now commonly known as Fuß-Catalan numbers (cf. [1, pp. 59-60] ).  ...

History of Catalan numbers [article]

Igor Pak
2014 arXiv   pre-print
This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.  ...  We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times.  ...  Catalan number.  ...
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