相关论文: Pascal arrays: counting Catalan sets
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…
We introduce the super Patalan numbers, a generalization of the super Catalan numbers in the sense of Gessel, and prove a number of properties analagous to those of the super Catalan numbers. The super Patalan numbers generalize the super…
The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we…
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain…
A coding method using binary sequences is presented for different computation problems related to Catalan numbers. This method proves in a very easy way the equivalence of these problems.
In the paper, the authors analytically generalize the Catalan numbers in combinatorial number theory, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy's integral formula in the…
A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial…
We demonstrate that a natural construction based on the two notions of insertion of a strand and finding the preimages of Temperley-Lieb algebra generators give an inductive means to generate all link patterns of a given number of strands.…
The main purpose of this note is to provide an elementary discussion of some simple triangles of integer numbers in particular through their connections with representation theory of $sl_2$. The triangles under consideration are the Catalan…
In this note we introduce several instructive examples of bijections found between several different combinatorially defined sequences of sets. Each sequence has cardinalities given by the Catalan numbers. Our results answer some questions…
This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula…
We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the…
Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work,…
We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in…
We introduce the three-Catalan triangle, highlighting the three-Catalan numbers along with their recurrence relation and combinatorial interpretation, which allows us to establish their log-convexity. Additionally, we prove that the rows of…
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving…
In this article, we use the Touchard identity in order to obtain new integral representations for Catalan numbers. The main idea consists in combining the identity with a known integral representation and resorting to the binomial theorem.…
We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that…
In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a…