The sorting index

TL;DR

This paper introduces the sorting index, a new Mahonian statistic for permutations, providing combinatorial descriptions and generalizations for Coxeter groups of types B and D, expanding understanding of permutation statistics.

Contribution

It defines and explores the sorting index, a novel Mahonian statistic, with detailed combinatorial descriptions and generalizations to Coxeter groups of types B and D.

Findings

Defined the sorting index and its combinatorial interpretation.

Extended the concept to type B and D Coxeter groups.

Connected the sorting index to length and reflection length generating functions.

Abstract

We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian statistic, which we call the sorting index. The sorting index of a permutation and its type B and type D analogues have natural combinatorial descriptions which we describe in detail.