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2009, ISBN 9780195334548, xiv, 422
Like the intriguing Fibonacci and Lucas numbers, Catalan numbers are also ubiquitous. "They have the same delightful propensity for popping up unexpectedly,... 
Catalan numbers (Mathematics) | Mathematics | Euler's triangulation problem | Pascal's triangle | Pascal's identity | Lucas numbers | Catalan sequence | Parenthesization problem | Catalan numbers | Martin gardner | Fibonacci numbers
Book
03/2015, ISBN 9781107075092, 225
Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their... 
Catalan numbers (Mathematics) | Matrices | Combinatorial analysis
eBook
Discrete Mathematics, ISSN 0012-365X, 11/2018, Volume 341, Issue 11, pp. 3159 - 3164
Three summation formulae on the λ-extended Catalan numbers are established by means of hypergeometric series approach with one of them being provided a... 
Reflection principle | Lattice path counting | Catalan numbers | MATHEMATICS
Journal Article
Indian Journal of Pure and Applied Mathematics, ISSN 0019-5588, 3/2019, Volume 50, Issue 1, pp. 181 - 189
We present certain q-analogues of Franel numbers and Catalan numbers, and establish several congruences on these q-numbers 
q -Catalan numbers | q -Franel numbers | Numerical Analysis | Mathematics, general | Mathematics | Applications of Mathematics | q -congruence | MATHEMATICS | q-Franel numbers | q-congruence | q-Catalan numbers | SUMS | Numbers | Research | Congruences and residues | Mathematical research
Journal Article
European Journal of Combinatorics, ISSN 0195-6698, 01/2017, Volume 59, pp. 114 - 121
The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers. In this paper, we give a combinatorial proof for a recurrence relation of... 
MATHEMATICS | ENUMERATION | CORE PARTITIONS | CATALAN NUMBERS | Partitions | Enumeration | Corals | Combinatorial analysis
Journal Article
Discrete Mathematics, ISSN 0012-365X, 10/2017, Volume 340, Issue 10, pp. 2388 - 2397
In this paper, we consider combinatorial numbers (Cm,k)m≥1,k≥0, mentioned as Catalan triangle numbers where Cm,k≔m−1k−m−1k−1. These numbers unify the entries... 
Combinatorial identities | Catalan numbers | Catalan triangle | Binomial coefficients | MATHEMATICS | IDENTITIES | SUMMATION | HARMONIC NUMBERS
Journal Article
Journal of Combinatorial Theory, Series A, ISSN 0097-3165, 01/2013, Volume 120, Issue 1, pp. 49 - 63
J. Piontkowski described the homology of the Jacobi factor of a plane curve singularity with one Puiseux pair. We discuss the combinatorial structure of his... 
Compactified Jacobians | [formula omitted]-Catalan numbers | Hilbert schemes | Q, t-Catalan numbers | MATHEMATICS | HILBERT SCHEME | STATISTICS | HOMOLOGY | CURVES | q, t-Catalan numbers
Journal Article
Publicationes Mathematicae, ISSN 0033-3883, 2017, Volume 90, Issue 3-4, pp. 387 - 406
Let {P-n}be the Catalan-Larcombe French numbers given by P-0 = 1, P-1 = 8 and n(2) P-n = 8(3n(2) - 3n+1)Pn-1-128(n-1)(2) Pn-2 (n >= 2), and let S-n = P-n/2(n).... 
Catalan-Larcombe-French number | Congruence | MATHEMATICS | congruence
Journal Article
Annals of Combinatorics, ISSN 0218-0006, 12/2018, Volume 22, Issue 4, pp. 819 - 874
Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the... 
Weyl group | Nilpotent orbit | Kreweras number | Catalan number | Mathematics | Reflection | Combinatorics | 51F15 | Narayana number | Cyclic sieving phenomenon | 20F55 | MATHEMATICS, APPLIED | REPRESENTATIONS | GREEN POLYNOMIALS | NONCROSSING PARTITIONS | COXETER GROUPS
Journal Article
TAIWANESE JOURNAL OF MATHEMATICS, ISSN 1027-5487, 02/2019, Volume 23, Issue 1, pp. 11 - 27
We prove that, for all positive integers n(1), ..., n(m), n(m+1) = n(1), and non-negative integers j and r with j <= m, the following two expressions 1/[n(1) +... 
MATHEMATICS | q-binomial coefficients | Q-ANALOGS | q-super Catalan numbers | cyclotomic polynomial | q-ballot numbers | CONGRUENCES | q-Catalan numbers
Journal Article
Mathematics, ISSN 2227-7390, 11/2018, Volume 6, Issue 12, p. 277
In the paper, the authors express the Fuss-Catalan numbers as several forms in terms of the Catalan-Qi function, find some analytic properties, including the... 
Logarithmic convexity | Fuss-Catalan number | Catalan-Qi function | Catalan number | Minimality | Complete monotonicity | Monotonicity | Inequality | GAMMA | Catalan-Qi-function | INEQUALITIES | REPRESENTATION | RATIO | monotonicity | complete monotonicity | logarithmic convexity | MATHEMATICS | inequality | VALUES | minimality | Fuss–Catalan number | Catalan–Qi function
Journal Article
Discrete Mathematics, ISSN 0012-365X, 02/2016, Volume 339, Issue 2, pp. 484 - 488
We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting... 
Recursive matrix | Hankel matrix | Catalan-like number | Riordan array | Totally positive matrix | Stieltjes moment sequence | MATHEMATICS | RIORDAN ARRAYS | Mathematics - Combinatorics
Journal Article
Economics Letters, ISSN 0165-1765, 11/2018, Volume 172, pp. 69 - 73
This paper shows that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction with k≥3 can... 
The revenue equivalence principle | [formula omitted]th price auction | Jensen’s identity | Catalan numbers | Hagen–Rothe’s identity | Jensen's identity | kth price auction | Hagen–Rothe's identity | IDENTITIES | COMPLETE INFORMATION | ECONOMICS | Hagen-Rothe's identity | Legislators | Auctions | Statistics
Journal Article
Advances in Applied Mathematics, ISSN 0196-8858, 05/2019, Volume 106, pp. 57 - 95
The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q,t)-Catalan numbers along with their corresponding γ-expansions... 
Gamma expansions | Pattern avoidance | [formula omitted]-Phenomenon | Catalan numbers | Narayana polynomials | Alternating permutations | (−1)-Phenomenon | MATHEMATICS, APPLIED | PERMUTATIONS | (-1)-Phenomenon | EULERIAN POLYNOMIALS | Combinatorics | Mathematics
Journal Article
SpringerPlus, ISSN 2193-1801, 12/2016, Volume 5, Issue 1, pp. 1 - 20
Journal Article
Discrete Mathematics, ISSN 0012-365X, 04/2017, Volume 340, Issue 4, pp. 708 - 712
The Schröder number Sn counts the number of the lattice paths from (0,0) to (n,n), containing no point above the line y=x and using only steps (1,0), (0,1) and... 
Schröder number | Congruence | MATHEMATICS | Schroder number | MOTZKIN NUMBERS | CATALAN | MODULO POWERS | Cytokinins
Journal Article
Discrete Mathematics, ISSN 0012-365X, 07/2019, Volume 342, Issue 7, pp. 2139 - 2147
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... 
Catalan number sequence | Stieltjes determinate sequences | Bernstein functions | Fuss–Catalan number sequence | Stieltjes moment sequences | Double factorial number sequence | MATHEMATICS | HAUSDORFF | DENSITIES | Fuss-Catalan number sequence | Mathematics - Combinatorics
Journal Article
Discrete Mathematics, ISSN 0012-365X, 06/2017, Volume 340, Issue 6, pp. 1389 - 1396
Let (a0,a1,a2,…) be the sequence of Catalan-like numbers. We evaluate the Hankel determinants of the shifted sequence (0,a0,a1,a2,…). As an application, we... 
Somos sequence | Catalan-like number | Hankel determinant | MATHEMATICS | SERIES REVERSION | SEQUENCES | TRANSFORM | SOMOS-4 | LAURENT PHENOMENON
Journal Article
Annals of Combinatorics, ISSN 0218-0006, 3/2016, Volume 20, Issue 1, pp. 21 - 58
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... 
diagonal harmonics | 05E10 | q, t -Catalan numbers | rational Catalan numbers | Shuffle Conjecture | 05E05 | rational parking functions | Mathematics | 05E18 | Combinatorics | q, t-Catalan numbers | MATHEMATICS, APPLIED | COMPACTIFIED JACOBIANS | STATISTICS | CONJECTURES | COMBINATORIAL FORMULA | CONTINUOUS FAMILY | CHARACTER
Journal Article
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