  In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function
$\operatorname{spt}_\omega(n)$, which is closely related to Ramanujan's third
order mock theta function $\omega(q)$. Garvan and Jennings-Shaffer introduce a
crank function which explains congruences for $\operatorname{spt}_\omega(n)$.
In this note, we study asymptotic behavior of this crank function and confirm a
positivity conjecture of the crank asymptotically. We also study a sign pattern
of the crank and congruences for $\operatorname{spt}_\omega(n)$.
