Counting Formulae for Square-tiled Surfaces in Genus Two

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 \pi$ each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for enumeration of square-tiled surfaces in these four diagrams, completing the detailed count for genus two. The formulae are in terms of various well-studied arithmetic functions, enabling us to give asymptotics for each diagram using a new calculation for additive convolutions of divisor functions that was recently derived by the author and collaborators. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.