Commutation classes of reduced decompositions of the longest element of the Weyl group with one element 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?
 A: I believe that for $n \geq 4$ there will be exactly $4$ such reduced words. One such word, call it $R_n$ can be constructed by starting with $s_{n-1}s_{n-2} \cdots s_2s_1s_2 \cdots s_{n-2}s_{n-1}$ and appending $R_{n-1}$ with all indices shifted up by 1. For example, $$R_6=s_5s_4s_3s_2s_1s_2s_3s_4s_5s_4s_3s_2s_3s_4s_3.$$
Three more words can be obtained by replacing each $s_i$ with $s_{n-i}$, reversing the word, or doing both.
For $n=4,5,6$ these are the only possibilities, and this probably shouldn't be too hard to argue in general.
A: The reduced words that are in their own commutation classes are:

*

*$s = [123\cdots(n-2)(n-1)(n-2)\cdots321][23\cdots(n-3)(n-2)(n-3)\cdots32][3\cdots(n-4)(n-3)(n-4)\cdots3]\cdots$

*the reversal of $s$ (writing the word in reverse order)

*the complement of $s$ (mapping each letter $i$ to $n-i$)

*the reverse complement of $s$
This is just one word in $S_2$ and two words in $S_3$ (no big surprises there!). Otherwise it is four words. For example, in $S_6$, they are

*

*123454321234323

*323432123454321

*543212345432343

*343234543212345

On a separate note, to add to the list of related topics, you might want to look at Fishel et al. (https://doi.org/10.1016/j.ejc.2018.07.002) about how commutation classes and braid classes interact.
