  Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an
analysis of the probability that two overlapping consecutive $k$-permutations
are order isomorphic, the authors of a recent paper showed that the expected
number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$
in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact
that random permutations pack consecutive patterns near-perfectly. We use
entirely different methods, namely the Stein-Chen method of Poisson
approximation, to reprove and slightly improve their result.
