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We present a procedure to sample uniformly from the set of combinatorialisomorphism types of balanced triangulations of surfaces - also known asgraph-encoded surfaces. For a given number $n$, the sample is a weighted set ofgraph-encoded surfaces with $2n$ triangles.  The sampling procedure relies on connections between graph-encoded surfacesand permutations, and basic properties of the symmetric group.  We implement our method and present a number of experimental findings basedon the analysis of $138$ million runs of our sampling procedure, producinggraph-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 theassociated genus distribution more and more concentrates on a vanishing portionof all possible genera as $n$ tends to infinity. Finally, we observe from ourdata that the mean number of non-trivial symmetries of a uniformly chosen graphencoding of a surface decays to zero at a rate super-exponential in $n$.

A Uniform Sampling Procedure for Abstract Triangulations of Surfaces