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

TL;DR

This paper explores the distribution of Mahonian and Euler-Mahonian statistics over permutations and partitions, revealing connections to $q$-Stirling numbers and the Delta Conjecture through bijections and interpretations.

Contribution

It introduces new bijective proofs and interpretations linking Mahonian statistics, $q$-Stirling numbers, and the Delta Conjecture, extending to ordered partitions and multiset partitions.

Findings

Distribution of BAST matches major index on pattern-avoiding permutations

Alternative interpretations of $q$-Stirling numbers are provided

Connections established between Mahonian statistics and the Delta Conjecture

Abstract

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.