Catalan Numbers (Selected Chapters of Number Theory: Special Numbers)

Catalan numbers, named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894), arise in a variety of combinatorial problems. They have many interesting properties, a rich history, and numerous arithmetic, number-theoretical, analytical, and combinatorial connections, as well as a variety of classical and modern applications. Considering the long list of open problems and questions related to the classical case, its relatives (Bell numbers, Motzkin numbers, Narayana numbers, etc.) and its generalizations, this book provides a broad perspective on the theory of this class of special numbers that will be of interest to professionals, students, and a general audience. The book begins with the history of the problem, before defining the considered numerical sets. The recurrence equation, closed formula, and generating function are then presented, followed by the simplest properties and number-theoretical properties. Later chapters discuss the relationships between Catalan numbers and other special numbers, as well as their applications and open problems.

Elena (Ivanovna) Deza is a Full Professor at Moscow Pedagogical State University, where she has been since 2006. Her research focuses on Analytic Number Theory, Discrete Mathematics, Theory of Discrete Metric Spaces, and Problems of Mathematical Education. She has written over 50 research papers, several mathematical textbooks and books on the teaching of mathematics in Russian, and several books/monographs in English, including Figurate Numbers (2012), Mersenne Numbers and Fermat Numbers (2021), Perfect and Amicable Numbers (2023), and Stirling Numbers (2024). Her mathematical teaching publications are mainly published in the following leading Russian journals: Mathematics in School, Kvant, and Vestnik of University of Russian Academy of Education. One of her special courses at Moscow State Pedagogical and Moscow Independent Universities is "Topics in Special Numbers". Deza has also published 11 mathematical textbooks in Russian.