How permutations displace points and stretch intervals

TL;DR

This paper investigates how permutations displace points and stretch intervals, analyzing average displacement, maximal displacement, and the extremal permutations that achieve these maxima, revealing precise formulas and characterizations.

Contribution

It provides exact formulas for maximal displacement measures in permutations and characterizes all permutations that attain these maxima.

Findings

Expected displacement ratio approaches 1/3 as n grows large.

Explicit formulas for maximum average and geometric stretch measures.

Complete characterization of permutations with maximal displacement and stretch.

Abstract

Let $S_n$ be the set of permutations on $\{1,\,\dots,\,n\}$ and $\pi\in S_n$. Let $\mathrm{d}(\pi)$ be the arithmetic average of $\{|i-\pi(i)|;\;1\le i\le n\}$. Then $\mathrm{d}(\pi)/n\in[0,\,1/2]$, the expected value of $\mathrm{d}(\pi)/n$ approaches $1/3$ as $n$ approaches infinity, and $\mathrm{d}(\pi)/n$ is close to $1/3$ for most permutations. We describe all permutations $\pi$ with maximal $\mathrm{d}(\pi)$. Let $\mathrm{s}^+(\pi)$ and $\mathrm{s}^*(\pi)$ be the arithmetic and geometric averages of $\{|\pi(i)-\pi(i+1)|;\;1\le i<n\}$, and let $M^+$, $M^*$ be the maxima of $\mathrm{s}^+$ and $\mathrm{s}^*$ over $S_n$, respectively. Then $M^+=(2m^2-1)/(2m-1)$ when $n=2m$, $M^+ = (2m^2+2m-1)/(2m)$ when $n=2m+1$, $M^* = (m^m(m+1)^{m-1})^{1/(n-1)}$ when $n=2m$, and, interestingly, $M^* = (m^m(m+1)(m+2)^{m-1})^{1/(n-1)}$ when $n=2m+1>1$. We describe all permutations $\pi$, $\sigma$ with maximal $\mathrm{s}^+(\pi)$ and $\mathrm{s}^*(\sigma)$.