  Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class
group lattice points intersecting a closed ball of radius $R$ in
Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of
the Teichm\"{u}ller space. In contrast we show the number of Dehn twist lattice
points intersecting a closed ball of radius $R$ is coarsely asymptotic to
$e^{\frac{h}{2}R}$. Moreover, we show the number multi-twist lattice points
intersecting a closed ball of radius $R$ grows coarsely at least at the rate of
$R \cdot e^{\frac{h}{2}R}$.
