  We study the graph on reduced words with edges given by the Coxeter relations
for the symmetric group. We define a metric on reduced words for a given
permutation, analogous to Coxeter length for permutations, for which the graph
becomes ranked with unique maximal element. We show this metric extends
naturally to balanced tableaux, and use it to recover enumerative results of
Edelman and Greene and of Reiner and Roichman.
