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An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation
Discrete mathematics & theoretical computer science, Permutation Patterns 2023, Tome 26 (2024) no. 1.

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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.
DOI : 10.46298/dmtcs.12458
Classification : 05A05 
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Godbole, Anant; Swickheimer, Hannah. An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation. Discrete mathematics & theoretical computer science, Permutation Patterns 2023, Tome 26 (2024) no. 1. doi : 10.46298/dmtcs.12458. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12458/

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