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,\ldots,...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2024-05-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | http://dmtcs.episciences.org/12458/pdf |
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| Summary: | 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. |
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| ISSN: | 1365-8050 |