  Two permutations $(x_1,\dots,x_w)$ and $(y_1,\dots,y_w)$ are weakly similar
if $x_i<x_{i+1}$ if and only if $y_i<y_{i+1}$ for all $1\leqslant i \leqslant
w$. Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$ and let
$wt(\pi)$ denote the largest integer $w$ such that $\pi$ contains a pair of
disjoint weakly similar sub-permutations (called weak twins) of length $w$.
Finally, let $wt(n)$ denote the minimum of $wt(\pi)$ over all permutations
$\pi$ of $[n]$. Clearly, $wt(n)\le n/2$. In this paper we show that $\tfrac
n{12}\le wt(n)\le\tfrac n2-\Omega(n^{1/3})$.
  We also study a variant of this problem. Let us say that
$\pi'=(\pi(i_1),...,\pi(i_j))$, $i_1<\cdots<i_j$, is an alternating (or
up-and-down) sub-permutation of $\pi$ if $\pi(i_1)>\pi(i_2)<\pi(i_3)>...$ or
$\pi(i_1)<\pi(i_2)>\pi(i_3)<...$. Let $\Pi_n$ be a random permutation selected
uniformly from all $n!$ permutations of $[n]$. It is known that the length of a
longest alternating permutation in $\Pi_n$ is asymptotically almost surely
(a.a.s.) close to $2n/3$. We study the maximum length $\alpha(n)$ of a pair of
disjoint alternating sub-permutations in $\Pi_n$ and show that there are two
constants $1/3<c_1<c_2<1/2$ such that a.a.s. $c_1n\le \alpha(n)\le c_2n$.
  In addition, we show that the alternating shape is the most popular among all
permutations of a given length.
