Catalan numbers and lattice paths

摘要

We define Catalan numbers as the sequence of numbers correspondingto the number of triangulations of a convex polygon. The Catalannumbers appear in various mathematical contexts and there are manyother combinatorial interpretations of these numbers as well. In thisoverview firstly we present basic properties and the Catalan convolution.We describe fundamental interpretations, which are those forwhich the Catalan convolution can be easily seen or there is a simplecorrespondence with some of the other interpretations. We enumeratesome notable families of lattice paths. In particular, we show two familiesof Dyck paths with constraint on the step (1, a??1). Finally, we presentthe beautiful Nicholsa?? bijection between Shapiro and Whitworthpaths.