  We study three aspects of commutation classes of reduced decompositions: the
number of commutation classes, the structures of their corresponding graphs,
and the enumeration of subnetworks, a concept recently introduced by Warrington
[21]. Our bound for the number of commutation classes generalizes the works of
Knuth[12], Green and Losonczy [7], and Tenner [19]. We analyze the structure of
the graph G(w) using pattern avoidance, which provides an application of
Tenner's characterization of vexillary permutations in [19]. We also discuss
some connections between our work and recent developments in the strong Bruhat
order and the higher Bruhat order.
