Patterns in random permutations avoiding the pattern 132

Abstract

We consider a random permutation drawn from the set of 132-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after scaling by $n^{\lambda (\sigma )/2}$ where $\lambda (\sigma )$ is the length of $\sigma$ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

32 pages

References in corpus (5)

• Patterns in random permutations avoiding the pattern 132
• The copies of any permutation pattern are asymptotically normal
• The enumeration of permutations avoiding 2143 and 4231
• Bijections from Dyck paths to 321-avoiding permutations revisited
• Fixed points of 321-avoiding permutations

Cited by in corpus (17)

• Patterns in random permutations avoiding the pattern 132
• Universal limits of substitution-closed permutation classes
• Local convergence for permutations and local limits for uniform $\rho$-avoiding permutations with $|\rho |=3$
• Square permutations are typically rectangular
• Fixed points of 321-avoiding permutations
• Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels
• Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences
• Random Permutations -- A geometric point of view
• Pattern-avoiding permutations and Brownian excursion Part I: Shapes and fluctuations
• Scaling limits of permutations avoiding long decreasing sequences
• Asymptotic distribution of fixed points of pattern-avoiding involutions
• Prolific Compositions
• A View From The Bridge Spanning Combinatorics and Probability
• Moments of permutation statistics and central limit theorems
• Arcsine laws for random walks generated from random permutations with applications to genomics
• Classical pattern distributions in ${\mathcal{S}}_n(132)$ and ${\mathcal{S}}_n(123)$
• Bounded affine permutations II. Avoidance of decreasing patterns