We look at extensions of formulas given by Jovovic and recently proved by Dhar on integer partitions where the smallest part occurs at least $m$ times and on integer partitions with fixed differences between the largest and smallest parts where the smallest part occurs at least $k$ times. Our results extend Dhar's results for the $m=2$ and $k=1$ cases to the general cases for arbitrary $m$ and $k$. We also look at analogous results for overpartitions and $\ell$-regular partitions.