  We prove that the expected number of braid moves in the commutation class of
the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1
s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one.
This is a variant of a similar result by V. Reiner, who proved that the
expected number of braid moves in a random reduced word for the long element is
one. The proof is bijective and uses X. Viennot's theory of heaps and variants
of the promotion operator. In addition, we provide a refinement of this result
on orbits under the action of even and odd promotion operators. This gives an
example of a homomesy for a nonabelian (dihedral) group that is not induced by
an abelian subgroup. Our techniques extend to more general posets and to other
statistics.
