  In 2007, Andrews and Paule introduced the broken $k$-diamond partition
function $\Delta_{k}(n)$, which has received a lot of researches on the
arithmetic propertises. In this paper, we will prove the broken $k$-diamond
partition function satisfies the Laguerre inequalities of order $2$ and the
determinantal inequalities of order $3$ for $k=1$ or $2$. Moreover, we
conjectured the thresholds for the Laguerre inequalities of order $m$ and the
positivity of $m$-order determinants for $4\leq m\leq 14$ for the broken
$k$-diamond partition function when $k=1$ or $2$.

Combinatorics