  The distribution of certain Mahonian statistic (called $\mathrm{BAST}$)
introduced by Babson and Steingr\'{i}msson over the set of permutations that
avoid vincular pattern $1\underline{32}$, is shown bijectively to match the
distribution of major index over the same set. This new layer of
equidistribution is then applied to give alternative interpretations of two
related $q$-Stirling numbers of the second kind, studied by Carlitz and Gould.
Moreover, extensions to an Euler-Mahonian statistic over ordered set
partitions, and to statistics over ordered multiset partitions present
themselves naturally. The latter of which is shown to be related to the
recently proven Delta Conjecture. During the course, a refined relation between
$\mathrm{BAST}$ and its reverse complement $\mathrm{STAT}$ is derived as well.
