Catalan Matrix Research Articles

Schröder–Catalan Matrix and Compactness of Matrix Operators on Its Associated Sequence Spaces

In this article, the regular Schröder–Catalan matrix is constructed and acquired by benefiting Schröder and Catalan numbers. After that, two sequence spaces are introduced, described as the domain of Schröder–Catalan matrix. Additionally, some algebraic and topological properties of the spaces in question, such as completeness, inclusion relations, basis and duals, are examined. In the last two sections, the necessary and sufficient conditions of some matrix classes and compact operators related aforementioned spaces are presented.

Implementation of the self-sufficiency matrix (SSM) to support diagnosing people with complex social needs at the Social Services of Catalonia

AbstractThe Department of Social Rights of the Generalitat de Catalunya considered using a unique tool to identify people with complex social needs at Social Services centres and to support the diagnosis processes. After conducting a thorough search and selection process for various tools, the self-sufficiency matrix (SSM), a Dutch tool, was ultimately chosen.The tool was adapted to the Catalan context through a transcultural translation process, which included a pilot and validation process. This resulted in the creation of the Catalan matrix (SSM-CAT).A comprehensive implementation program was defined to start the adoption of the tool at basic social services. This program included training trainers and providing online training with practical cases. The implementation process was accompanied by support and monitoring to ensure success.Through this process, over 3,468 professionals (including 334 trainers) received training on the self-sufficiency matrix, and 31,354 individuals who received basic social services in Catalonia were evaluated. In Barcelona, a more thorough monitoring of the implementation was conducted, assessing a representative sample of the care provided (6,916 individuals attended) generating a more accurate description of the situation of the people attended by social services in the city of Barcelona.Disclosure of InterestNone Declared

Enumeration of anti-invariant subspaces and Touchard's formula for the entries of the q-Hermite Catalan matrix

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

On ideal convergence of sequences via Catalan matrix operator

Catalan sequence initially introduced by Euler but named after Eugene Catalan has found abundant applications in the fields of probability theory, number theory, analysis, combinatorics etc. Recently İlkhan formulated the matrix associated with Catalan numbers and studied its domain in c0(C~) and c(C~). In this paper, by employing Catalan matrix defined by Catalan sequence and convergence via ideal, we establish Catalan sequence spaces c0I(C~),cI(C~) and ℓ∞I(C~), study the properties of these corresponding spaces and establish inclusion relations.

Quaternion-Type Catalan Transforms of the ρ-Fibonacci and ρ-Lucas Numbers

In this paper, we define a new sequence called the quaternion-type Catalan sequence and give generating function, exponential representation, quaternionic Catalan matrix and its some properties. Then, we introduce a quaternion-type Catalan transform, which is a generalization of the Catalan transform, and its inverse. Moreover, we apply the quaternion-type Catalan transforms to ρ -Fibonacci and ρ -Lucas sequences and derive some new sequences by using these quaternion-type Catalan transforms.

Fermat's and Catalan's equations over $ M_2(\mathbb{Z}) $

<abstract><p>Let $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right) $ be a given matrix such that $ bc\neq0 $ and let $ C(A) = \{B\in M_2(\mathbb{Z}): AB = BA\} $. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $ uX^i+vY^j = wZ^k, \, i, \, j, \, k\in\mathbb{N}, \, X, \, Y, \, Z\in C(A) $, where $ u, \, v, \, w $ are given nonzero integers such that $ \gcd\left(u, \, v, \, w\right) = 1 $. From this, we get a necessary and sufficient condition for the solvability of the Fermat's matrix equation in $ C(A) $. Moreover, we show that the solvability of the Catalan's matrix equation in $ M_2\left(\mathbb{Z}\right) $ can be reduced to the solvability of the Catalan's matrix equation in $ C(A) $, and finally to the solvability of the Catalan's equation in quadratic fields.</p></abstract>

A study on q-analogue of Catalan sequence spaces

In this study, we construct q-analog C(q) of Catalan matrix and study the sequence spaces c0(C(q)) and c(C(q)) defined as the domain of q-Catalan matrix C(q) in the spaces c0 and c, respectively. We exhibit some topological properties, obtain Schauder bases and determine ?-, ?-, and ?-duals of the spaces c0(C(q)) and c(C(q)). Finally, we characterize certain class of matrix mappings from the spaces c0(C(q)) and c(C(q)) to the space ? = {??, c0, c, ?1} and give the necessary and sufficient conditions for a matrix operator to be compact.

Statistics of partial permutations via Catalan matrices

A generalized Catalan matrix (an,k)n,k≥0 is generated by two seed sequences s=(s0,s1,…) and t=(t1,t2,…) together with a recurrence relation. By taking sℓ=2ℓ+1 and tℓ=ℓ2 we can interpret an,k as the number of partial permutations, which are n×n0,1-matrices of k zero rows with at most one 1 in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on two interesting permutation families, namely the connected permutations and cycle-up-down permutations, are also given.

A New Approach to k-Jacobsthal Lucas Sequences

In this study, 〖CS〗_(k,n) of S_(k,n) Catalan transformation of 𝑘−Jacobsthal-Lucas sequences is defined. S_(k,n) Catalan transformation of 𝑘−Jacobsthal-Lucas S_(k,n) sequences is obtained.In addition the transformation of CS_(k,n) is written as the product of the Catalan matrix C, which is the lower triangular matrix, and the S_k matrix of type 𝑛 𝑥 1, and the Hankel transformations of some k-Jacobsthal-Lucas numbers has been found.

Some new observations on Catalan almost convergent sequence spaces and the Catalan core

In this article, we introduce and study Catalan almost convergent sequence spaces by using the Catalan matrix. We obtain topological properties and the β-dual of the resulting sequence spaces. Finally, we define the Catalan core of complex-valued sequences and give some inclusion theorems related to the Catalan core.

The Cyclic Groups Generated by Catalan Matrices

Let $n$ a be positive integer. The Catalan matrix of order $n$, denoted by $\mathscr{C}_n[x]$, is a lower triangular Toeplitz matrix whose nonzero elements are expressions containing Catalan numbers and a real parameter $x$. In this paper, we investigate the multiplicative order of the cyclic group generated by $\mathscr{C}_n[x]$ (for $x\in\mathbb{N}$) modulo a positive integer $m$.

Shevirat Ha-Kelim: Jewish Mysticism and the Catalan Matrix for Dialogue and Conflict

AbstractLate Medieval anti-Jewish violence is a well-known phenomenon, but its origins and institutionalization are still blurred and enigmatic. In thirteenth and fourteenth century Catalonia, the denouement of the increasing popular hostility against the Jewry was particularly dramatic. The seeds of violence were the result of a long and complex process of social, theological and political interactions. In this contribution, we will discuss the intellectual matrix of medieval anti-Semitism in Catalonia and its relationship with the rising of scholastics and with the theoretical foundations of Catalan politics. We will also approach its counterpart: the Jewish response to collective suffering.

A wide class of Combinatorial matrices related with Reciprocal Pascal and Super Catalan matrices

In this paper, we present a number of combinatorial matrices that are generalizations or variants of the super Catalan matrix and the reciprocal Pascal matrix. We present explicit formulae for LU-decompositions of all the matrices and their inverses. To prove the claimed results, we mainly use the celebrated Zeilberger algorithm.

Two Variants of the Reciprocal Super Catalan Matrix

In this paper, we define two kinds variants of the super Catalan matrix as well as their q-analoques. We give explicit expressions for LU-decompositions of these matrices and their inverses.

The generalized reciprocal super Catalan matrix

The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two additional parameters. Explicit formulae are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the q-version of Zeilberger's celebrated algorithm.

The reciprocal super Catalan matrix

Abstract The reciprocal super Catalan matrix has entries . Explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, q-analogues are also presented.

A variant of the reciprocal super Catalan matrix

Abstract Recently Prodinger [8] considered the reciprocal super Catalan matrix and gave explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and obtained some related matrices. For all results, q-analogues were also presented. In this paper, we define and study a variant of the reciprocal super Catalan matrix with two additional parameters. Explicit formulæ for its LU-decomposition, LUdecomposition of its inverse and the Cholesky decomposition are obtained. For all results, q-analogues are also presented.

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.

Ballot matrix as Catalan matrix power and related identities

We use an analytical approach to find the k th power of the Catalan matrix. Precisely, it is proven that the power of the Catalan matrix is a lower triangular Toeplitz matrix which contains the well-known ballot numbers. A result from [H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990, Free download available from http://www.math.upenn.edu/~wilf/Downld.html.], related to the generating function for Catalan numbers, is extended to the negative integers. Three interesting representations for Catalan numbers by means of the binomial coefficients and the hypergeometric functions are obtained using relations between Catalan matrix powers.

Some identities on Catalan numbers and hypergeometric functions via Catalan matrix power

In this paper we use the Catalan matrix power as a tool for deriving identities involving Catalan numbers and hypergeometric functions. For that purpose, we extend earlier investigated relations between the Catalan matrix and the Pascal matrix by inserting the Catalan matrix power and particulary the squared Catalan matrix in those relations. We also pay attention to some relations between Catalan matrix powers of different degrees, which allows us to derive the simplification formula for hypergeometric function 3 F 2, as well as the simplification formula for the product of the Catalan number and the hypergeometric function 3 F 2. Some identities involving Catalan numbers, proved by the non-matrix approach, are also given.