From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

Abstract

The distribution of certain Mahonian statistic (called BAST) introduced by Babson and Steingrímsson over the set of permutations that avoid vincular pattern 132_, 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 Delta Conjecture. During the course, a refined relation between BAST and its reverse complement STAT is derived as well.