Equidistributions of Mahonian Statistics over Pattern Avoiding Permutations

Abstract

A Mahonian $d$-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern statistics of length at most $d$. Babson and Steingrímsson classified all Mahonian 3-functions up to trivial bijections and identified many of them with well-known Mahonian statistics in the literature. We prove a host of Mahonian 3-function equidistributions over pattern avoiding sets of permutations. Tools used include block decomposition, Dyck paths and generating functions.