How permutations displace points and stretch intervals

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 $\mathrm{s}^*(\sigma)$.