Os números combinatórios de Catalan

Abstract

Present in problems of combinatorics, algebra, computer science, graph theory and geometry, Catalan numbers are little known in Brazil and have little literature in the Portuguese language. With this perspective, from this work we try to contribute with the insertion of bibliography on the theme and to encourage and spread the study of it. We present a complete exploration of Catalan numbers, exposing their history, including a brief biography of Belgian mathematician Eug`ene Charles Catalan. We display and demonstrate properties involving algebraic and geometric aspects of the numerical sequence. We describe some applications such as lattice paths and polygon triangulations. We relate Catalan numbers with other historical numbers and arrangements, such as Pascal’s pi and triangle. Initially, we bring a summary of binomial theory and other important results for the presentation and definition of Catalan numbers. Finally, we propose a didactic sequence with a theoretical framework and a suggested practice that can be applied by teachers in the classroom to explore Catalan numbers.