Lehmer code transforms and Mahonian statistics on permutations

TL;DR

This paper presents an alternative proof that certain Mahonian permutation statistics, expressible via vincular patterns, can be characterized through new permutation codes derived from Lehmer's code.

Contribution

It introduces new code transforms based on Lehmer's code that generate permutation codes counting vincular pattern occurrences, providing an alternative proof of Mahonian properties.

Findings

New code transforms relate Lehmer's code to vincular pattern counts

Alternative proof of Mahonian permutation statistics

Permutation codes can be used to analyze pattern occurrences

Abstract

In 2000 Babson and Steingr{\'\i}msson introduced the notion of vincular patterns in permutations. They shown that essentially all well-known Mahonian permutation statistics can be written as combinations of such patterns. Also, they proved and conjectured that other combinations of vincular patterns are still Mahonian. These conjectures were proved later: by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In this paper we give an alternative proof of some of these results. Our approach is based on permutation codes which, like Lehmer's code, map bijectively permutations onto subexcedant sequences. More precisely, we give several code transforms (i.e., bijections between subexcedant sequences) which when applied to Lehmer's code yield new permutation codes which count occurrences of some vincular patterns.