The central component of a triangulation

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.

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