A classification of mahonian maj-inv statistics

TL;DR

This paper classifies Mahonian statistics on words that can be expressed as sums of generalized inversion and major index statistics, extending classical results and providing a comprehensive characterization.

Contribution

It provides a complete characterization of Mahonian statistics formed as sums of generalized inversion and major index statistics.

Findings

Identifies all Mahonian statistics expressible as sums of generalized statistics

Extends classical Mahonian results to broader classes of statistics

Provides a framework for analyzing Mahonian properties in generalized settings

Abstract

Two well known mahonian statistics on words are the inversion number and the major index. In 1996, Foata and Zeilberger introduced generalizations, parameterized by relations, of these statistics. In this paper, we study the statistics which can be written as a sum of these generalized statistics. This leads to generalizations of some classical results. In particular, we characterize all such statistics which are mahonian.