  Triangulations are important objects of study in combinatorics, finite
element simulations and quantum gravity, where its entropy is crucial for many
physical properties. Due to their inherent complex topological structure even
the number of possible triangulations is unknown for large systems. We present
a novel algorithm for an approximate enumeration which is based on calculations
of the density of states using the Wang-Landau flat histogram sampling. For
triangulations on two-dimensional integer lattices we achive excellent
agreement with known exact numbers of small triangulations as well as an
improvement of analytical calculated asymptotics. The entropy density is
$C=2.196(3)$ consistent with rigorous upper and lower bounds. The presented
numerical scheme can easily be applied to other counting and optimization
problems.

Computational Physics