Rational generating series for affine permutation pattern avoidance

TL;DR

This paper studies affine permutations avoiding fixed patterns, providing rational generating series representations, characterizations of pattern periodicity, and a new polyhedral encoding for the affine symmetric group.

Contribution

It introduces a rational generating series for affine pattern-avoiding permutations and characterizes when coefficients are periodic, using a novel polyhedral encoding.

Findings

Generating series are always rational functions.

Characterization of patterns with periodic coefficients.

A new polyhedral encoding for the affine symmetric group.

Abstract

We consider the set of affine permutations that avoid a fixed permutation pattern. Crites has given a simple characterization for when this set is infinite. We find the generating series for this set using the Coxeter length statistic and prove that it can always be represented as a rational function. We also give a characterization of the patterns for which the coefficients of the generating series are periodic. The proofs exploit a new polyhedral encoding for the affine symmetric group.