Enumerating pattern avoidance for affine permutations

TL;DR

This paper investigates pattern avoidance in affine permutations, establishing finiteness conditions related to the pattern 321 and counting affine permutations avoiding patterns in S_3, with conjectures for S_4.

Contribution

It characterizes when affine permutations avoiding a pattern are finite and provides enumeration results for patterns in S_3, along with conjectures for S_4.

Findings

Finitely many affine permutations avoid p iff p avoids 321.

Counted affine permutations avoiding patterns in S_3.

Proposed conjectures for patterns in S_4.

Abstract

In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern p, there are only finitely many affine permutations in $\widetilde{S}_n$ that avoid p if and only if p avoids the pattern 321. We then count the number of affine permutations that avoid a given pattern p for each p in S_3, as well as give some conjectures for the patterns in S_4.