  We give a bijective proof of Macdonald's reduced word identity using pipe
dreams and Little's bumping algorithm. This proof extends to a principal
specialization due to Fomin and Stanley. Such a proof has been sought for over
20 years. Our bijective tools also allow us to solve a problem posed by Fomin
and Kirillov from 1997 using work of Wachs, Lenart, Serrano and Stump. These
results extend earlier work by the third author on a Markov process for reduced
words of the longest permutation.
