Enumeration and Wilf-classification of permutations avoiding five patterns of length 4

doi:10.47443/cm.2020.0006

Abstract

Highlights

A permutation on the set [n] = {1, 2, . . . , n} is any arrangement of the elements of [n]
For two permutations τ = τ1τ2 · · · τk ∈ Sk and σ = σ1σ2 · · · σn ∈ Sn, we say that τ occurs as a pattern in σ if there exists a subsequence in σ which is order-isomorphic to τ, that is, there exist k indices 1 ≤ i1 < i2 < · · · < ik ≤ n such that σia < σib if and only if τa < τb for all 1 ≤ a, b ≤ k
The research of pattern avoidance has received a lot of attention in the last couple of decades

Summary

Introduction

We denote the number of distinct Wilf classes for the permutations avoiding exactly k distinct patterns from S4 by wk. Vatter [29] showed that 12 of the 38 Wilf classes can be enumerated with so-called regular insertion encoding algorithm (the INSENC algorithm) This algorithm computes the (necessarily rational) generating function for any regular class, namely a class of permutations avoiding a set T of patterns that has a regular insertion encoding (see [1]). Some of these generating functions were computed by hand by Kremer and Shiu [10].

Enumeration and Wilf-classification of permutations avoiding five patterns

Conclusion