Generalized Catalan Numbers from Hypergraphs

The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.

On Coloring Catalan Number Distance Graphs and Interference Graphs

A vertex coloring of a graph G is a mapping that allots colors to the vertices of G. Such a coloring is said to be a proper vertex coloring if two vertices joined by an edge receive different colors. The chromatic number χ ( G ) is the least number of colors used in a proper vertex coloring. In this paper, we compute the χ of certain distance graphs whose distance set elements are (a) a finite set of Catalan numbers, (b) a finite set of generalized Catalan numbers, (c) a finite set of Hankel transform of a transformed sequence of Catalan numbers. Then while discussing the importance of minimizing interference in wireless networks, we probe how a vertex coloring problem is related to minimizing vertex collisions and signal clashes of the associated interference graph. Then when investigating the χ of certain G ( V , D ) and graphs with interference, we also compute certain lower and upper bound for χ of any given simple graph in terms of the average degree and Laplacian operator. Besides obtaining some interesting results we also raised some open problems.

Engenharia Didática de Formação (EDF): sobre o ensino dos Números (Generalizados) de Catalan (NGC)Didactical Engineering: about the teaching of generalized Catalan numbers

Registramos uma considerável atenção dedicada por parte dos autores de livros de História da Matemática (HM) concernentemente aos clássicos fundamentos do Cálculo Diferencial e Integral. Por outro lado, se mostra imprescindível ao entendimento do professor de Matemática uma compreensão sobre um irrefreável processo matemático e epistemológico evolutivo dos objetos matemáticos, desde seu estádio de nascedouro até o momento atual. Assim, o presente trabalho relata uma Engenharia Didática de Formação (EDF) desenvolvida com a participação de cinco professores em formação inicial, no Instituto Federal de Educação, Ciência e Tecnologia – IFCE, no ano de 2017. O tema abordado envolveu a noção de Números Generalizados de Catalan (NGC) que representa uma contribuição de vários matemáticos e a pesquisa atual sobre inúmeros problemas derivados confirmam seu processo de generalização ininterrupto. O estudo envolveu cinco tarefas e duas situações estruturadas de ensino, com o aporte da Teoria das Situações Didáticas (TSD). Os dados coligidos evidenciam várias propriedades e, sobretudo, teoremas e definições matemáticas descobertas e formuladas pelos sujeitos participantes da investigação o que concorreu para o incremento de suas habilidades profissionais e um conhecimento histórico, epistêmico e pragmático sobre a noção. We have recorded considerable attention on the part of the authors of Mathematical History (MH) books concerning the classical fundamentals of Differential and Integral Calculus. On the other hand, it is essential to the understanding of the Mathematics teacher an understanding about an unstoppable mathematical and evolutionary epistemological process of the mathematical objects, from its nascent stage to the present moment. Thus, the present work reports a Training Didactic Engineering (EDF) developed with the participation of five teachers in initial formation, in the Federal Institute of Education, Science and Technology - IFCE, in the year 2017. The topic covered involved the notion of Numbers Generalized Catalan (NGC) that represents a contribution of several mathematicians and the current research on numerous derived problems confirm its process of uninterrupted generalization. The study involved five tasks and two structured teaching situations, with the contribution of the Theory of Educational Situations (TSD). The collected data show several properties and, above all, the theorems and mathematical definitions discovered and formulated by the subjects participating in the research, which contributed to the increase of their professional skills and a historical, epistemic and pragmatic knowledge about the notion.

CUNTZ–KRIEGER ALGEBRAS AND A GENERALIZATION OF CATALAN NUMBERS

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra [Formula: see text] for the transition matrix AGof the directed edges of G. The generalized Catalan numbers [Formula: see text] enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.