Orbit cardinality formulas for primitive square-tiled surfaces in H(2)

Let $n$ be an odd integer, and consider the two orbits of primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$, denoted orbit A and orbit B. For a prime $p$, write $\prod_{p\mid n}(1-1/p^2)$ for the product over prime divisors of $n$. Orbit cardinality conjecture. For odd $n$, the cardinalities of the two orbits are

$$\frac{3}{16}(n-1)n^2\prod_{p\mid n}\left(1-\frac{1}{p^2}\right)$$

for orbit A and

$$\frac{3}{16}(n-3)n^2\prod_{p\mid n}\left(1-\frac{1}{p^2}\right)$$

for orbit B. These formulas refine the known formula for the total number of primitive square-tiled surfaces and agree with the leading term established by the paper's counting theorem for prime $n$; their general validity is supported by numerical evidence.