  The minimal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is the
smallest positive integer that is not a part of $\lambda$. For a positive
integer $n$, $ \sigma\, \rm{mex}(n)$ denotes the sum of the minimal excludants
of all partitions of $n$. Recently, Andrews and Newman obtained a new
combinatorial interpretations for $\sigma\, \rm{mex}(n)$. They showed, using
generating functions, that $\sigma\, \rm{mex}(n)$ equals the number of
partitions of $n$ into distinct parts using two colors.
  In this paper, we provide a purely combinatorial proof of this result and new
properties of the function $\sigma\, \rm{mex}(n)$. We generalize this
combinatorial interpretation to $\sigma_r\, \rm{mex}(n)$, the sum of least
$r$-gaps in all partitions of $n$. The least $r$-gap of a partition $\lambda$
is the smallest positive integer that does not appear at least $r$ times as a
part of $\lambda$.
