Affine permutations and an affine Catalan monoid

Abstract

We describe results on pattern avoidance arising from the affine Catalan monoid. The schema of affine codes as canonical decompositions in conjunction with two-row moves is detailed, and then applied in studying the Catalan quotient of the 0-Hecke monoid. We prove a conjecture of Hanusa and Jones concerning periodicity in the number of fully-commutative affine permutations. We then re-frame prior results on fully commutative elements using the affine codes.