Links and the Diaconis-Graham Inequality

TL;DR

This paper reveals a deep connection between permutation measures of disarray, topological links, and the Euler characteristic, extending previous characterizations and providing new insights into permutation structure.

Contribution

It extends Woo's topological link characterization to all permutations, relating discrepancy in inequalities to the Euler characteristic of associated links.

Findings

Discrepancy relates to the Euler characteristic of associated links.

Characterization of permutations with fixed discrepancy via links.

Stabilized-interval-free permutations correspond to nonsplit links.

Abstract

In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in $\mathbb R^3$ that can be associated to the cycle diagram of a permutation. We show that Woo's characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham's inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.