Graphical Mahonian Statistics on Words

TL;DR

This paper extends the theory of graphical permutation statistics by characterizing when certain statistics are equidistributed on words, introducing a new graphical sorting index, and analyzing their distribution properties.

Contribution

It generalizes classical permutation statistics to words via graphical indices and characterizes graphs for which these statistics are equidistributed on a single rearrangement class.

Findings

Equidistribution of graphical statistics characterizes bipartitional graphs.

Introduction of a graphical sorting index generalizing the permutation sorting index.

Characterization of graphs where the sorting index is equidistributed with other graphical statistics.

Abstract

Foata and Zeilberger defined the graphical major index, $\mathrm{maj}'_U$, and the graphical inversion index, $\mathrm{inv}'_U$, for words. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and $\mathrm{inv}$ indexed by directed graphs $U$. They showed that $\mathrm{maj}'_U$ and $\mathrm{inv}'_U$ are equidistributed over all rearrangement classes if and only if $U$ is bipartitional. In this paper we strengthen their result by showing that if $\mathrm{maj}'_U$ and $\mathrm{inv}'_U$ are equidistributed on a single rearrangement class then $U$ is essentially bipartitional. Moreover, we define a graphical sorting index, $\mathrm{sor}'_U$, which generalizes the sorting index of a permutation. We then characterize the graphs $U$ for which $\mathrm{sor}'_U$ is equidistributed with $\mathrm{inv}'_U$ and $\mathrm{maj}'_U$ on a single rearrangement class.