We prove that the expected number of braid moves in the commutation class of the 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 reduced word of the long element is one. The proof uses Viennot's theory of heaps and variants of the promotion operator.

This is joint work with a working group at a recent AIM workshop, including Zach Hamaker, Nicolas Thiery, Graham White, and Nathan Williams.