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 $\mathrm{ s}^*(\sigma)$.
Categories: math.CO