  In a recent pioneering work, Andrews and Newman defined an extended function
$p_{A,a}(n)$ of their minimal excludant or "mex" of a partition function. By
considering the special cases $p_{k,k}(n)$ and $p_{2k,k}(n)$, they unearthed
connections to the rank and crank of partitions and some restricted partitions.
In this paper, we build on their work and obtain more general results
associating the extended mex function with the number of partitions of an
integer with arbitrary bound on the rank and crank. We also derive a new result
expressing the smallest parts function of Andrews as a finite sum of the
extended mex function in consideration with a curious coefficient. We also
obtain a few restricted partition identities with some reminiscent of shifted
partition identities. Finally, we define and explore a new minimal excludant
for overpartitions.
