相关论文: Pascal arrays: counting Catalan sets

We describe a basic correspondence between linear algebraic structures within vector embeddings in artificial neural networks and conditional independence constraints on the probability distributions modeled by these networks. Our framework…

A logic calculus is presented that is a conservative extension of linear logic. The motivation beneath this work concerns lazy evaluation, true concurrency and interferences in proof search. The calculus includes two new connectives to deal…

Collatz Conjecture sequences increase and decrease in seemingly random fashion. By identifying and analyzing the forms of numbers, we discover that Collatz sequences are governed by very specific, well-defined rules, which we call cascades.

In this paper, we generalize the Catalan number to the $(n,k)$-th Catalan numbers and find a combinatorial description that the $(n,k)$-th Catalan numbers is equal to the number of partitions of $n(k-1)+2$ polygon by $(k+1)$-gon where all…

We reprove a few results concerning paperfolding sequences using properties of Catalan numbers modulo 2.

In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.

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…

In this paper we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix - a special matrix which has only the natural numbers as entries and is closely…

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…

In this note, we provide a bijection between a new collection of words on nonnegative integers of length n and Dyck paths of length 2n-2, thus proving that this collection belongs to the Catalan family. The surprising key step in this…

We give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters.

Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are…

This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…

Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain…

A generalized Catalan matrix $(a_{n,k})_{n,k\ge 0}$ is generated by two seed sequences $\mathbf{s}=(s_0,s_1,\ldots)$ and $\mathbf{t}=(t_1,t_2,\ldots)$ together with a recurrence relation. By taking $s_\ell=2\ell+1$ and $t_\ell=\ell^2$ we…

In this work we introduce declarative statistics, a suite of declarative modelling tools for statistical analysis. Statistical constraints represent the key building block of declarative statistics. First, we introduce a range of relevant…

Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…

In this paper, we study Catalan numbers which can be represented by the p-adic integral on Zp and we investigate some properties and formulae related to Catalan numbers and special numbers.

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner…

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums…