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 [2] showed that the expected number of distinct consecutive patterns of all lengths k is an element of {1, 2, ... , n} in pi n is n22 (1 - o(1)) as n -> infinity. 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.