Generalised Pattern Avoidance

TL;DR

This paper studies generalized permutation pattern avoidance, providing formulas for permutations avoiding certain patterns, and explores connections to combinatorial structures like set partitions and paths.

Contribution

It offers a complete solution for counting permutations avoiding single length-three patterns with one adjacent pair and introduces monotone partitions linked to non-overlapping partitions.

Findings

Formulas for permutations avoiding specific patterns

Connections to Dyck paths, Motzkin paths, and involutions

Introduction of monotone partitions and their properties

Abstract

Recently, Babson and Steingrimsson have introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters. We also give some results for the number of permutations avoiding two different patterns. Relations are exhibited to several well studied combinatorial structures, such as set partitions, Dyck paths, Motzkin paths, and involutions. Furthermore, a new class of set partitions, called monotone partitions, is defined and shown to be in one-to-one correspondence with non-overlapping partitions.