Geodesic
Pattern Avoidance for Random Permutations
Discrete mathematics & theoretical computer science, Permutation Patterns 2016, Tome 19 (2017-2018) no. 2.

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Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences and a corresponding joint distribution of independent Bernoulli random variables, which as a corollary yields a Poisson approximation for the distribution of the number of occurrences of any pattern. We also investigate occurrences of consecutive patterns in random Mallows permutations, of which uniform random permutations are a special case. These bounds allow us to estimate the probability that a pattern occurs any number of times and, in particular, the probability that a random permutation avoids a given pattern. 
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     author = {Crane, Harry and DeSalvo, Stephen},
     title = {Pattern {Avoidance} for {Random} {Permutations}},
     journal = {Discrete mathematics & theoretical computer science},
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Crane, Harry; DeSalvo, Stephen. Pattern Avoidance for Random Permutations. Discrete mathematics & theoretical computer science, Permutation Patterns 2016, Tome 19 (2017-2018) no. 2. doi : 10.23638/DMTCS-19-2-13. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-2-13/

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