Variations on Catalan Connections and the Children Pairing Game
摘要
Catalan numbers are “dense” in combinatorics with hundreds of interpretations in combinatorics, algebra, geometry, and topology. This lecture starts with a simple interpretation of Catalan numbers with a topological flavor and presents a proof, motivated by the theory of skein modules, but which is very elementary and looks like a child’s game. We also discuss the lattice path and Dyck path interpretation of Catalan numbers (including Désiré André reflection principle). Furthermore we introduce Fuss numbers, a generalization of Catalan numbers by Euler’s assistant at St. Petersburg.