Filters








46,303 Hits in 1.8 sec

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 [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.  ... 
doi:10.1016/0097-3165(87)90033-1 fatcat:fw6ripqq5fdudck5speiyrnb24

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.  ... 
doi:10.1016/j.laa.2012.10.001 fatcat:yawaorcdiza3dfuda4otedxaji

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.  ... 
doi:10.1016/0097-3165(94)90119-8 fatcat:ubp52f7ki5gxdnipmnlybi33aq

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.  ... 
doi:10.1002/9781118159743.ch35 fatcat:h2mge2own5h5dex733dhjxkd3u

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?  ... 
arXiv:1101.1108v1 fatcat:okjw3c4mxjbmhibsrfpior2cma

Noncommutative Catalan numbers [article]

Arkady Berenstein, Vladimir Retakh
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.  ... 
arXiv:1708.03316v4 fatcat:56mnub6vprea5ot56jjj7z7dpe

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.  ... 
doi:10.1016/0097-3165(85)90089-5 fatcat:4opq75sohfglroy5quipd74i24

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.  ... 
doi:10.37236/674 fatcat:lxcaegnsincbrh6o6q6cbahiam

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.  ... 
arXiv:2011.09593v1 fatcat:trt6l7hosncnpahacxocx3g73a

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.  ... 
arXiv:2104.05701v1 fatcat:zhwdlgx6ffbfxcf3prk233waoy

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 .  ... 
arXiv:1508.01688v5 fatcat:jbaujxt5mndfdlsluj72qxeooe

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.  ... 
doi:10.1016/0022-247x(85)90212-4 fatcat:ddllpqcyczhgtaohb4uyw6yun4

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.  ... 
doi:10.4064/bc96-0-15 fatcat:dn2do2dk2zg3rfeiorytxs4334

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] ).  ... 
arXiv:1101.1416v2 fatcat:7ycqvl5tyjff5m736fgyklioyi

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.  ... 
arXiv:1408.5711v2 fatcat:wugoedmr5jhvnm6i7ocvlfxde4
« Previous Showing results 1 — 15 out of 46,303 results