  Topological triangulations of orientable and non-orientable surfaces with
arbitrary genus have important applications in quantum geometry, graph theory
and statistical physics. However, until now only the asymptotics for 2-spheres
are known analytically, and exact counts of triangulations are only available
for both small genus and small triangulations. We apply the Wang-Landau
algorithm to calculate the number $N(m,h)$ of triangulations for several order
of magnitudes in system size $m$ and genus $h$. We verify that the limit of the
entropy density of triangulations is independent of genus and orientability and
are able to determine the next-to-leading and the next-to-next-to-leading order
terms. We conjecture for the number of surface triangulations the asymptotic
behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h -
1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide
a mathematicians proof for the exact asymptotics.

Computational Physics;High Energy Physics - Theory;Combinatorics