  We present a procedure to sample uniformly from the set of combinatorial
isomorphism types of balanced triangulations of surfaces - also known as
graph-encoded surfaces. For a given number $n$, the sample is a weighted set of
graph-encoded surfaces with $2n$ triangles.
  The sampling procedure relies on connections between graph-encoded surfaces
and permutations, and basic properties of the symmetric group.
  We implement our method and present a number of experimental findings based
on the analysis of $138$ million runs of our sampling procedure, producing
graph-encoded surfaces with up to $280$ triangles.
  Namely, we determine that, for $n$ fixed, the empirical mean genus
$\bar{g}(n)$ of our sample is very close to $\bar{g}(n) = \frac{n-1}{2} -
(16.98n -110.61)^{1/4}$. Moreover, we present experimental evidence that the
associated genus distribution more and more concentrates on a vanishing portion
of all possible genera as $n$ tends to infinity. Finally, we observe from our
data that the mean number of non-trivial symmetries of a uniformly chosen graph
encoding of a surface decays to zero at a rate super-exponential in $n$.
