  Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$
consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$
consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on
$\mathbb{N}$ with $p_j>0$. Let $P_n$ denote the probability measure on $S_n$
corresponding to the $p$-shifted random permutation. Our main result, under the
additional assumption that $\{p_j\}_{j=1}^\infty$ is non-increasing, is that $$
\begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n
)}_{l,k})=\big(\prod_{j=1}^{k-1}\sum_{i=1}^jp_i\big)
\big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big), \end{aligned} $$ and that if
$\lim_{n\to\infty}\min(k_n,n-k_n)=\infty$, then $$ \begin{aligned}
&\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k_n})=
\big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big)^2. \end{aligned} $$ In particular
these limits are positive if and only if $\sum_{j=1}^\infty jp_j<\infty$. We
say that super-clustering occurs when the limits are positive. We also give a
new characterization of the class of $p$-shifted probability distributions on
$S_\infty$.

Probability;Combinatorics