How permutations displace points and stretch intervals

Abstract

Let $S_n$ be the set of permutations on $\{1,\,\dots,\,n\}$ and $π\in S_n$. Let $\mathrm{d}(π)$ be the arithmetic average of $\{|i-π(i)|;\;1\le i\le n\}$. Then $\mathrm{d}(π)/n\in[0,\,1/2]$, the expected value of $\mathrm{d}(π)/n$ approaches $1/3$ as $n$ approaches infinity, and $\mathrm{d}(π)/n$ is close to $1/3$ for most permutations. We describe all permutations $π$ with maximal $\mathrm{d}(π)$. Let $\mathrm{s}^+(π)$ and $\mathrm{s}^*(π)$ be the arithmetic and geometric averages of $\{|π(i)-π(i+1)|;\;1\le i 1$. We describe all permutations $π$, $σ$ with maximal $\mathrm{s}^+(π)$ and $\mathrm{s}^*(σ)$.