
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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Bayesian Methods and Mixture Models · Advanced Mathematical Identities
