Generalized Catalan Numbers and Some Divisibility Properties

Abstract

Generalized Catalan Numbers and Some Divisibility Properties by Jacob Bobrowski Dr. Peter Shiue, Examination Committee Chair Professor of Mathematical Sciences University of Nevada, Las Vegas I investigate the divisibility properties of generalized Catalan numbers by extending known results for ordinary Catalan numbers to their general case. First, I define the general Catalan numbers and provide a new derivation of a known formula. Second, I show several combinatorial representations of generalized Catalan numbers and survey bijections across these representation. Third, I extend several divisibility results proved by Koshy. Finally, I prove conditions under which sufficiently large primes form blocks of divisibility and indivisibility of the generalized Catalan numbers, extending a known result by Alter and Kubota.