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

Let πₙ 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∈\1,2,…,n\ in πₙ is n²/2(1-o(1)) as n→∞. 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.