  We develop the technique of reduced word manipulation to give a range of
results concerning reduced words and permutations more generally. We prove a
broad connection between pattern containment and reduced words, which
specializes to our previous work for vexillary permutations. We also analyze
general tilings of Elnitsky's polygon, and demonstrate that these are closely
related to the patterns in a permutation. Building on previous work for
commutation classes, we show that reduced word enumeration is monotonically
increasing with respect to pattern containment. Finally, we give several
applications of this work. We show that a permutation and a pattern have
equally many reduced words if and only if they have the same length
(equivalently, the same number of 21-patterns), and that they have equally many
commutation classes if and only if they have the same number of 321-patterns.
We also apply our techniques to enumeration problems of pattern avoidance, and
give a bijection between 132-avoiding permutations of a given length and
partitions of that same size, as well as refinements of this data and a
connection to the Catalan numbers.
