On spt-crank type functions

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function ${\mathrm{spt}}_{\omega }(n)$, 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 ${\mathrm{spt}}_{\omega }(n)$. 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 ${\mathrm{spt}}_{\omega }(n)$.