An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation

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 $k\in \{1,2,\dots ,n\}$ in ${\pi }_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to \mathrm{\infty }$. 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.