The Generalized Catalan Numbers
The study of the Catalan numbers dates back to Euler, and there are literally hundreds of realizations of the Catalan numbers throughout combinatorics. They are related to the symmetric group, geometry, and even have applications to organic chemistry. One definition of the Catalan number C_n is the number of ways to triangulate a regular n-gon with non-crossing diagonals. We are studying the generalized Catalan numbers, which count the number of ways to dissect a polygon into smaller regular polygons (rather than just triangles), and different realizations of them. In particular, we are interested in the number of orbits of these objects under the natural action of the dihedral group, and a characterization of the orbits. We are also interested in how these orbits manifest themselves in other settings.