Reduced Decompositions and Permutation Patterns

Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. We generalize that result with a study of permutations via their reduced decompositions and the notion of pattern containment. These techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a particular poset. Additionally, the combined frameworks yield several new results about the commutation classes of a permutation. In particular, these describe structural aspects of the corresponding graph of the classes and the zonotopal tilings of a polygon defined by Elnitsky that is associated with the permutation.