Pattern Avoidance for Random Permutations

Abstract

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.

24 pages, 2 Figures, 4 Tables

References in corpus (5)

• Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns
• A new upper bound for 1324-avoiding permutations
• Simultaneous avoidance of generalized patterns
• The enumeration of permutations avoiding 2143 and 4231
• Approaches for enumerating permutations with a prescribed number of occurrences of patterns

Cited by in corpus (2)

• A Central Limit Theorem for Vincular Permutation Patterns
• A central limit theorem for descents of a Mallows permutation and its inverse