Sequence Of Catalan Numbers Research Articles

Sequences from Fibonacci to Catalan: A combinatorial interpretationviaDyck paths

We use Dyck paths having some restrictions in order to give a combinatorial interpretation for some famous number sequences. Starting from the Fibonacci numbers we show how thek-generalized Fibonacci numbers, the powers of 2, the Pell numbers, thek-generalized Pell numbers and the even-indexed Fibonacci numbers can be obtained by means of constraints on the number of consecutive valleys (at a given height) of the Dyck paths. By acting on the maximum height of the paths we get a succession of number sequences whose limit is the sequence of Catalan numbers. For these numbers we obtain a family of interesting relations including afull historyrecurrence relation. The whole study can be accomplished also by involving particular sets of stringsviaa simple encoding of Dyck paths.

Thin set theorems and cone avoidance

The thin set theorem R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty ,\ell } asserts the existence, for every k k -coloring of the subsets of natural numbers of size n n , of an infinite set of natural numbers, all of whose subsets of size n n use at most ℓ \ell colors. Whenever ℓ = 1 \ell = 1 , the statement corresponds to Ramsey’s theorem. From a computational viewpoint, the thin set theorem admits a threshold phenomenon, in that whenever the number of colors ℓ \ell is sufficiently large with respect to the size n n of the tuples, the thin set theorem admits strong cone avoidance. Let d 0 , d 1 , … d_0, d_1, \dots be the sequence of Catalan numbers. For n ≥ 1 n \geq 1 , R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } admits strong cone avoidance if and only if ℓ ≥ d n \ell \geq d_n and cone avoidance if and only if ℓ ≥ d n − 1 \ell \geq d_{n-1} . We say that a set A A is R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } -encodable if there is an instance of R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } such that every solution computes A A . The R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } -encodable sets are precisely the hyperarithmetic sets if and only if ℓ > 2 n − 1 \ell > 2^{n-1} , the arithmetic sets if and only if 2 n − 1 ≤ ℓ > d n 2^{n-1} \leq \ell > d_n , and the computable sets if and only if d n ≤ ℓ d_n \leq \ell .

A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces c and c0

ABSTRACTThe main purpose of this paper is to define a new conservative matrix by means of the fascinating sequence of Catalan numbers and study the matrix domain of this newly introduced matrix in the classical sequence spaces c and . Additionally, after determining the Köthe-Toeplitz dual, generalized Köthe-Toeplitz dual and Garling dual, certain matrix transformations and compact operators are characterized on the new Banach spaces.

Supercombinator set acquired from context-free grammar samples

We present an algorithm that transforms context-free grammars into a non-redundant set of supercombinators. This set contains interconnected lambda calculus’ supercombinators that are enriched by grammar operations. The resulting set is scalable and it can be extended with new supercombinators created from grammars. We describe this algorithm in detail and then we apply it on 62,008 grammar samples in order to find out the properties and limits of acquired supercombinator set. We show that this set has a maximum theoretical limit of possible supercombinators. That limit is the sequence of Catalan numbers. We show that in some cases we are able to reach that limit if we use large enough input data source and we limit the size of supercombinators permitted into the final set. We also describe another benefit of our algorithm, which is the identification of most reoccurring structures in the input set.

Some Results on a Class of Polynomials Related to Convolutions of the Catalan Sequence

Motivated by the search of the singular values of Jordan blocks, in a previous paper (Capparelli and Maroscia in Med J Math 10:1609–1630, 2013) we studied, among other things, a family of monic polynomials with integer coefficients that turned out to be linked to convolutions of the sequence of Catalan numbers. In the present paper, we continue the study of these polynomials and prove, in particular, the irreducibility of an infinite subset of them. As an interesting byproduct, we also obtain a simple rational function in two variables which can be naturally thought of as the generating function of the Catalan number sequence and all its convolutions.

A direct derivation of the Catalan formula

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formulawith its alternative formThese can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

Generation and Enumeration of Some Classes of Interval Orders

In this paper we consider the class of interval orders, recently considered by several authors from both an algebraic and an enumerative point of view. According to Fishburn’s Theorem (Fishburn J Math Psychol 7:144–149, 1970), these objects can be characterized as posets avoiding the poset 2 + 2. We provide a recursive method for the unique generation of interval orders of size n + 1 from those of size n, extending the technique presented by El-Zahar (1989) and then re-obtain the enumeration of this class, as done in Bousquet-Melou et al. (2010). As a consequence we provide a method for the enumeration of several subclasses of interval orders, namely AV(2 + 2, N), AV(2 + 2, 3 + 1), AV(2 + 2, N, 3 + 1). In particular, we prove that the first two classes are enumerated by the sequence of Catalan numbers, and we establish a bijection between the two classes, based on the cardinalities of the principal ideals of the posets.

A family of eigensequences

We construct a family ( ν k ) k⩾1 of ‘eigensequences’ of a certain operator studied previously by Cameron (Discrete Math. 75 (1989) 89) and Bernstein and Sloane (Linear Algebra Appl. 228 (1–3) (1995) 57). Here ν 1 is the sequence of Catalan numbers and ν 2 is the sequence of little Schröder numbers.

The Favoured Solutions of the Moment Problem

The sequence of Catalan numbers is known to be the moments of the Wigner law. A characterization of this sequence is given in terms of some determinants and the moment sequence of the q-Gaussian law is characterized in a similar fashion.

Sur les Polynômes de Catalan Simples et Doubles

Two families of polynomials are introduced, which generalize the sequence of Catalan numbers. Both families are applied to the solution of a previously unsolved problem about characterization of tree types.

Pascal triangles, Catalan numbers and renewal arrays

In response to some recent questions of L.W. Shapiro, we develop a theory of triangular arrays, called renewal arrays, which have arithmetic properties similar to those of Pascal's triangle. The Lagrange inversion formula has an important place in this theory and there is a close relation between it and the theory of renewal sequences. By way of illustration, we give several examples of renewal arrays of combinatorial interest, including complete generalizations of the familiar Pascal triangle and sequence of Catalan numbers.

Prime and prime power divisibility of Catalan numbers

For any prime p, the sequence of Catalan numbers a n= 1 n 2n−2 n−1 is divided by the a n prime to p into blocks B k ( k > 0) of a n divisible by p. The lengths and positions of the B k are determined. Additional results are obtained on prime power divisibility of Catalan numbers.