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Catalan numbers, q-Catalan numbers and hypergeometric series

1987
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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 [4] 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. ...

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Parametric Catalan numbers and Catalan triangles

2013
*
Linear Algebra and its Applications
*

Among the generalized

doi:10.1016/j.laa.2012.10.001
fatcat:yawaorcdiza3dfuda4otedxaji
*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*. ...##
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Catalan numbers revisited

1994
*
Journal of combinatorial theory. Series A
*

THE

doi:10.1016/0097-3165(94)90119-8
fatcat:ubp52f7ki5gxdnipmnlybi33aq
*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. ...##
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Generalized Catalan Numbers
[chapter]

2011
*
Fibonacci and Catalan Numbers
*

In this paper we present an algebraic interpretation for generalized

doi:10.1002/9781118159743.ch35
fatcat:h2mge2own5h5dex733dhjxkd3u
*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. ...##
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Eulerian-Catalan Numbers
[article]

2011
*
arXiv
*
pre-print

We show that the Eulerian-

arXiv:1101.1108v1
fatcat:okjw3c4mxjbmhibsrfpior2cma
*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

arXiv:1708.03316v4
fatcat:56mnub6vprea5ot56jjj7z7dpe
*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*. ...##
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q-Catalan numbers

1985
*
Journal of combinatorial theory. Series A
*

most obvious q-analog of the

doi:10.1016/0097-3165(85)90089-5
fatcat:4opq75sohfglroy5quipd74i24
*Catalan**numbers*. ... They will be the starting point of our study of q-*Catalan**numbers*. ...##
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Eulerian-Catalan Numbers

2011
*
Electronic Journal of Combinatorics
*

We show that the Eulerian-

doi:10.37236/674
fatcat:lxcaegnsincbrh6o6q6cbahiam
*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]

2020
*
arXiv
*
pre-print

A finitization of the

arXiv:2011.09593v1
fatcat:trt6l7hosncnpahacxocx3g73a
*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*. ...##
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Positroid Catalan numbers
[article]

2021
*
arXiv
*
pre-print

Given a permutation $f$, we study the positroid

arXiv:2104.05701v1
fatcat:zhwdlgx6ffbfxcf3prk233waoy
*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*. ...##
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Modular Catalan Numbers
[article]

2016
*
arXiv
*
pre-print

The

arXiv:1508.01688v5
fatcat:jbaujxt5mndfdlsluj72qxeooe
*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

1985
*
Journal of Mathematical Analysis and Applications
*

Since the

doi:10.1016/0022-247x(85)90212-4
fatcat:ddllpqcyczhgtaohb4uyw6yun4
*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. ...##
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Remarks on Catalan and super-Catalan numbers

2011
*
Banach Center Publications
*

In this article we discuss the

doi:10.4064/bc96-0-15
fatcat:dn2do2dk2zg3rfeiorytxs4334
*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. ...##
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Chromatic statistics for Catalan and Fuß-Catalan numbers
[article]

2011
*
arXiv
*
pre-print

We refine

arXiv:1101.1416v2
fatcat:7ycqvl5tyjff5m736fgyklioyi
*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] ). ...##
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History of Catalan numbers
[article]

2014
*
arXiv
*
pre-print

This note will appear as an appendix in Richard Stanley's forthcoming book on

arXiv:1408.5711v2
fatcat:wugoedmr5jhvnm6i7ocvlfxde4
*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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