  We prove that permutations with few inversions exhibit a local-global
dichotomy in the following sense. Suppose ${\boldsymbol\sigma}$ is a
permutation chosen uniformly at random from the set of all permutations of
$[n]$ with exactly $m=m(n)\ll n^2$ inversions. If $i<j$ are chosen uniformly at
random from $[n]$, then ${\boldsymbol\sigma}(i)<{\boldsymbol\sigma}(j)$
asymptotically almost surely. However, if $i$ and $j$ are chosen so that
$j-i\ll m/n$, and $m \ll n^2/\log^2 n$, then
$\lim_{n\to\infty}\mathbb{P}\big[{\boldsymbol\sigma}(i)<{\boldsymbol\sigma}(j)\big]=\frac{1}{2}$.
Moreover, if $k=k(n)\ll \sqrt{m/n}$, then the restriction of
${\boldsymbol\sigma}$ to a random $k$-point interval is asymptotically
uniformly distributed over $\mathcal{S}_k$. Thus, knowledge of the local
structure of ${\boldsymbol\sigma}$ reveals nothing about its global form. We
establish that $\sqrt{m/n}$ is the threshold for local uniformity and $m/n$ the
threshold for inversions, and determine the behaviour in the critical windows.
  As pointed out by a referee, there are flaws in the proofs that do not seem
easily rectifiable (see comments on pages 9 and 15). So the results stated
above have not been established.

Combinatorics