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

From $q$-Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns