  The central component of a polygon triangulation is defined as the triangle
or diameter that contain its geometric center. More generally, every polygon
dissection contains a central component. Using this notion, we derive new
recurrences for the Catalan and $k$-Catalan numbers, and use these recursions
to prove congruence relations of these numbers. We also enumerate the
triangulations that contain a fixed vertex in their central components.

Combinatorics