Fix a set of nonzero positive integers $\{k_i\}$ none of which equal 6. How many compact oriented surface triangulations with 2n triangles are there whose vertices have valences $k_1, ..., k_n, 6, ..., 6$ ? I will discuss recent work proving that the answer is the nth Fourier coefficient of a quasi-modular form, using techniques from representation theory pioneered by Eskin and Okounkov. This generalizes joint work with Peter Smillie (which I will talk about in the postdoc seminar) showing that if one counts with the correct weight, the number of triangulations of the sphere with all $k_i<6$ is exactly $809/2612138803200 * \sigma_9(n)$ where $\sigma_9(n)$ is the sum of the ninth powers of the divisors of n.