On a conjecture of Andrews and Bachraoui

Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1\le k\le 4$. In this note, we prove that $F_{k,1}(q)$ has non-negative coefficients for $5\le k\le 10$. Moreover, we show that, as $k\to \mathrm{\infty }$, $F_{k,1}(q)$ is related to Ramanujan’s third order mock theta function $\omega (q)$ and to quotients of certain $q$-binomial coefficients.