Two-Dimensional Catalan Numbers

The *Catalan number, C(n),* is the number of ways a polygon with *n* sides can be triangulated with no additional vertices. These numbers can be calculated recursively. We generalize the problem to the next higher dimension. Given a triangulation *T* of the 2-sphere, define the *Catalan number of T *to be the number of non-degenerate ``triangulations'' of the ball with tetrahedra that have boundary isomorphic to *T *and which only use vertices in *T*.

**Question:** *How do you compute the Catalan number of a triangulation of the 2-sphere?*

The smallest example is the complete graph *K(4)*. This is trivial as there is only one such decomposition into tetrahedra. The next smallest example is *K(5)-K(2)*. The problem generalizes to even higher dimensions.

Submitted by: Steve Fisk, Department of Mathematics, Bowdoin College, Brunswick ME 04011-2599 USA.

Send comments to *dan.archdeacon@uvm.edu *

August, 1995