How permutations displace points and stretch intervals

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 ${|i-\pi(i)|;;1\le i\le n}$\mathrm{.}Then$$\mathrm{d}$(\pi)/n\in[0,,1/2]$,theexpectedvalueof$$\mathrm{d}$(\pi)/n$approaches$1/3$as$n$approachesinfinity,and$$\mathrm{d}$(\pi)/n$iscloseto$1/3$ for most permutations. We describe all permutations $\pi$ with maximal $\mathrm{d}(\pi )$. Let ${\mathrm{s}}^{+}(\pi )$ and ${\mathrm{s}}^{\ast }(\pi )$ be the arithmetic and geometric averages of $\{\mathrm{\mid }\pi (i)-\pi (i+1)\mathrm{\mid };1\le i<n\}$, and let $M^{+}$, $M^{\ast }$ be the maxima of ${\mathrm{s}}^{+}$ and ${\mathrm{s}}^{\ast }$ over $S_n$, respectively. Then $M^{+}=(2m^2-1)\mathrm{/}(2m-1)$ when $n=2m$, $M^{+}=(2m^2+2m-1)\mathrm{/}(2m)$ when $n=2m+1$, $M^{\ast }=(m^m(m+1{)}^{m-1}{)}^{1\mathrm{/}(n-1)}$ when $n=2m$, and, interestingly, $M^* = (m^m(m+1)(m+2)^{m-1})^{1/(n-1)}$when$n=2m+1>1$\mathrm{.}Wedescribeallpermutations$\pi$,$\sigma$withmaximal$$\mathrm{s}$^+(\pi)$and$$\mathrm{s}$^*(\sigma)$.