  Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a
property of their reduced decompositions. This paper generalizes that result
with a detailed 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.
