How permutations displace points and stretch intervals

Daniel Daly, Petr Vojtěchovský

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}(π)/n\in[…