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We show that the number of square-tiled surfaces of genus $g$, with $n$marked points, with one or both of its horizontal and vertical foliationsbelonging to fixed mapping class group orbits, and having at most $L$ squares,is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing inMirzakhani's count of simple closed hyperbolic geodesics. Many of the resultsin this paper reflect recent discoveries of Delecroix, Goujard, Zograf, andZorich, but the approach considered here is very different from theirs. Wefollow conceptual and geometric methods inspired by Mirzakhani's work.

Counting square-tiled surfaces with prescribed real and imaginary  foliations and connections to Mirzakhani's asymptotics for simple closed  hyperbolic geodesics