For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the number of such decompositions is discussed.

Following Stembridge, we define a relation $∼$ on the set of reduced expressions for $w_0$. Let $\mathbf{w}$ and $\mathbf{w}'$ be two reduced expressions for $w_0$ and define $\mathbf{w} \sim \mathbf{w}'$ if we can obtain $\mathbf{w}'$ from $\mathbf{w}$ by applying a single commutation. Now, define the equivalence relation $\simeq$ by taking the reflexive transitive closure of $\sim$. Each equivalence class under $\simeq$ is called a commutation class.

The question of the number of commutation classes (of reduced expressions for $w_0 \in S_n$) is still an open problem. However, is there an formula for the number of equivalence classes containing just one element?

expected number of braid movesthat one can apply to a random reduced expression for $w_0$: see Reiner (doi.org/10.1016/j.ejc.2004.06.010), Tenner (ajc.maths.uq.edu.au/pdf/62/ajc_v62_p147.pdf), Schilling et al. (doi.org/10.1016/j.ejc.2016.10.008) $\endgroup$