Cyclic Pattern Containment and Avoidance

TL;DR

This paper extends the study of pattern avoidance from linear to cyclic permutations, establishing a cyclic Erdős–Szekeres theorem, analyzing avoidance of multiple patterns, and exploring cyclic descent statistics.

Contribution

It introduces a cyclic variant of the Erdős–Szekeres theorem and advances understanding of pattern avoidance in cyclic permutations, including generating functions for cyclic descent.

Findings

Cyclic Erdős–Szekeres theorem established

Results on avoidance of multiple length-4 patterns

Generating functions for cyclic descent statistic derived

Abstract

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we establish a cyclic variant of the Erdos-Szekeres Theorem that any linear permutation of length mn+1 must contain either the increasing pattern of length m+1 or the decreasing pattern of length n+1. We then derive results about avoidance of multiple patterns of length 4. We also determine generating functions for the cyclic descent statistic on these classes. Finally, we end with various open questions and avenues for future research.