Related papers: A Generic Numbering System based on Catalan Famili…
In this note, we provide bijective proofs of some identities involving the Bell number, as previously requested. Our arguments may be extended to yield a generalization in terms of complete Bell polynomials. We also provide a further…
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…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
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…
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For…
We present an algorithmic mapping from permutations of length dn to labeled n-node d-ary trees and back again. Given such a bijection, one can interpret each of the factorials in the formula for the Catalan numbers as a count of…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling…
In this document, we propose a description, via a Haskell implementation, of a generalization of the notion of regular expression allowing us to group the definitions and the methods of (tree or word) automata constructions over one generic…
A Catalan magma is a unique factorisation normed magma with only one irreducible element. The \val partitions the base set into subsets enumerated by Catalan numbers. The primary theorem characterises the conditions which a set a with…
In this paper, we present an involution on some kind of colored $k$-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers $C_{k,\gamma}(n)=\frac{\gamma}{k n+\gamma}{k n+\gamma\choose…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows…
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…
Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show…
We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of…
We describe a family of new algorithms for finding the canonical image of a set of points under the action of a permutation group. This family of algorithms makes use of the orbit structure of the group, and a chain of subgroups of the…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.