  By an $r$-tuplet in a permutation we mean a family of $r$ pairwise disjoint
subsequences with the same relative order. The length of an $r$-tuplet is
defined as the length of any single subsequence in the family. Let $t^{(r)}(n)$
denote the largest $k$ such that every permutation of length $n$ contains an
$r$-tuplet of length $k$. We prove that $t^{(r)}(n)=O\left(n^{\frac
r{2r-1}}\right)$ and $t^{(r)}(n)=\Omega\left( n^{\frac{R}{2R-1}} \right)$,
where $R=\binom{2r-1}r$. We conjecture that the upper bound brings the correct
order of magnitude of $t^{(r)}(n)$ and support this conjecture by proving that
it holds for almost all permutations. Our work generalizes previous studies of
the case $r=2$.
