On a weighted generalization of Kendall's tau distance

TL;DR

This paper introduces a weighted Kendall's tau distance metric for permutations, explores its properties using permutohedron structures, and identifies conditions for specific metric space configurations.

Contribution

It presents a novel weighted generalization of Kendall's tau distance and analyzes its geometric and combinatorial properties.

Findings

Provides a criterion for permutation ordering within the metric space

Identifies conditions for pseudolinear quadruples in the metric space

Uses permutohedron edge graph to study permutation metrics

Abstract

We introduce a metric on the set of permutations of given order, which is a weighted generalization of Kendall's $\tau$ rank distance and study its properties. Using the edge graph of a permutohedron, we give a criterion which guarantees that a permutation lies metrically between another two fixed permutations. In addition, the conditions under which four points from the resulting metric space form a pseudolinear quadruple were found.