Patterns in random permutations avoiding the pattern 321

Abstract

We consider a random permutation drawn from the set of 321‐avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by nm + ℓ where m is the length of σ and ℓ is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.

Patterns in Random Permutations Avoiding Some Sets of Multiple Patterns

“…Let σ = 21. Then w 21 = 2 −1/2 , see [13], and thus (3.5)-(3.6), with ℓ = 1 and m 1 = m = 2, yield for the number of inversions,…”

“…There are 6 cases where a single permutation of length 3 is avoided, but by the symmetries in Subsection 2.2 these reduce to 2 non-equivalent cases, for example 132 (equivalent to 231, 213, 312) and 321 (equivalent to 123). These cases are treated in detail in [12] and [13], respectively. Both analyses are based on bijections with binary trees and Dyck paths, and the well-known convergence in distribution of random Dyck paths to a Brownian excursion, but the details are very different, and so are in general the resulting limit distributions.…”

“…For comparison with the results in later sections, we quote the main results of [12] and [13], referring to these papers for further details and proofs. Recall that the standard Brownian excursion e(x) is a random non-negative function on [0, 1].…”

“…There is also a number of papers by various authors that study other properties of random τ -avoiding permutations, see e.g. the references in [13]; such results will not be considered here.…”

Moments of permutation statistics and central limit theorems

“…The recent works of Gaetz and Ryba [30] and Kammoun [38] establish normal limit laws on certain classes of permutations for classical and vincular patterns, respectively. In addition, Janson [35,36] showed that the number of pattern occurrences is not normally distributed when we sample from the permutations avoiding a certain fixed pattern. Earlier, Janson, Nakamura and Zeilberger [37] initiated the study of the same general question.…”