Results 1 to 10 of about 7,617 (257)
Multivariate Fuss–Catalan numbers [PDF]
Discrete Mathematics, 2008 Catalan numbers $C(n)=\frac{1}{n+1}{2n\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of factors is given by ballot numbers $B(n,k)=\frac{n-k}{n+k}{n+k\choose n}$. These integers are known to satisfy simple recurrence, which may be visualised in a ``Catalan triangle'', a lower-triangular two-
Feng Qi, Xiao-Ting Shi, Pietro Cerone
exaly +6 more sources
Some properties of the Catalan-Qi function related to the Catalan numbers. [PDF]
Springerplus, 2016 In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan-Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically ...
Qi F, Mahmoud M, Shi XT, Liu FF.
europepmc +4 more sources
On the Catalan Numbers and Some of Their Identities [PDF]
Symmetry, 2019 The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the help of these sequences, we obtained the identities for the convolution involving the Catalan numbers.
Wenpeng Zhang
exaly +3 more sources
Eulerian-Catalan Numbers [PDF]
The Electronic Journal of Combinatorics, 2011 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.
Hoda Bidkhori, Seth Sullivant
openaire +3 more sources
Noncommutative Catalan Numbers [PDF]
Annals of Combinatorics, 2019 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 ...
Berenstein, A., Retakh, V.
openaire +3 more sources
The Complexity of Trees, Universal Grammar and Economy Conditions
Biolinguistics, 2022 In this squib, I argue that the child faces a severe computational complexity problem in parsing even the simplest of trees: the number of possible trees consistent with UG grows exponentially as a function of the number of lexical items.
Chris Collins
doaj +1 more source
New Expressions for Sums of Products of the Catalan Numbers
Axioms, 2021 In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan ...
Conghui Xie, Yuan He
doaj +1 more source
CFT correlators, W $$ \mathcal{W} $$ -algebras and generalized Catalan numbers
Journal of High Energy Physics, 2022 In two spacetime dimensions the Virasoro heavy-heavy-light-light (HHLL) vacuum block in a certain limit is governed by the Catalan numbers. The equation for their generating function can be generalized to a differential equation which the logarithm of ...
Robin Karlsson +4 more
doaj +1 more source
The dual of number sequences, Riordan polynomials, and Sheffer polynomials
Special Matrices, 2021 In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences.
He Tian-Xiao, Ramírez José L.
doaj +1 more source
Symmetry of Narayana Numbers and Rowvacuation of Root Posets
Forum of Mathematics, Sigma, 2021 For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains.
Colin Defant, Sam Hopkins
doaj +1 more source