Mahonian Statistics and Vincular Patterns on Permutations over Multisets

TL;DR

This paper explores Mahonian statistics expressed through vincular patterns on permutations with repetitions, identifying new Mahonian statistics extensions and proving their properties using involutions.

Contribution

It extends the method of vincular pattern combinations to discover and prove new Mahonian statistics on permutations with repetitions.

Findings

Identified 8 vincular-pattern combinations that are Mahonian.

Proved new Mahonian statistics extensions using involutions.

Extended existing Mahonian statistics to permutations with repetitions.

Abstract

Most Mahonian statistics can be expressed as a linear combination of vincular patterns. This is not only true with statistics on the permutation set, but it can also be applied for statistics on the permutation with repetition set. By following the method extending the vincular patterns combinations presented by Kitaev and Vajnovszki, we discover 8 vincular-patterns combinations of mad and madl extensions that are possible to be Mahonian. Some of these have been proved to be Mahonian on repetitive permutations by Clarke, Steingrimsson and Zeng, while the rest are new statistics extensions. In this thesis, we determine combinations of vincular pattern extension of mad and madl in Clarke, Steingrimsson and Zeng s paper, which have been proved to be Mahonian on the repetitive permutations. This result will be used to support the proof of Mahonity of the new statistics extensions. We show that these new statistics extensions are also Mahonian by constructing an involution {\Phi} on repetitive permutations, which preserves the descents statistics and transforms new statistics extensions to Mahonian mad and madl extensions of Clarke.