A certain kind of permutations and transport of Bruhat chains under conjugation

Let $$(W,S)$$ be a finite Coxeter system. Let us consider the following situation:

Let $$v_1,v_2,w\in W$$ such that $$v_1=wv_2w^{-1}$$. Let $$s_{\beta_r}\ldots s_{\beta_1}$$ be a reduced expression of $$v_2$$. We consider the condition C that $$1 is a saturated chain in the Bruhat order.

If C is satisfied for one reduced expression of $$v_2$$, it is satisfied for any. Further, we then must have $$\ell(v_1)=\ell(v_2)$$.

I want to consider solutions for the problem C. I know some, e.g., if $$w=1$$ and $$v_1=v_2$$ or if $$v_1=v_2=w=w_o$$. These are rather trivial. It is also possible to find solutions when $$v_1,v_2\in S$$ (or maybe even in $$T$$).

Let us consider elements $$\pi$$ such that $$\pi S\pi^{-1}\cap S=\emptyset$$.

Questions.

1. Do permutations $$\pi$$ as above have a name and where somewhere studied in the literature? Where?
2. Can problem C be solved for some non-trivial* $$v_1,v_2\neq 1$$ and some $$w$$ of the form $$w=v_1\pi$$ with $$\pi$$ as above? Has problem C been studied in the literature? Where?

*Footnote: There is also the trivial solution $$v_1=v_2=\pi^{-1}$$ which corresponds to $$w=1$$ listed as trivial above. I am interested if there are any more possible solutions for $$w$$ of this form.

Partial answers for the symmetric group are also very interesting...

• I think for the symmetric group at least, we have $\pi S \pi^{-1}=\varnothing$ if and only if the one line notation of $\pi$ has no two adjacent letters $i$ and $j$ with $|i-j|=1$. E.g., $\pi=13524$ satisfies this. Apr 19 '19 at 16:54
• Yes, this is true. I studied those for a longer time now, and if the number of letters grows, there are plenty. That's why I ask if there is something about it in the literature. Apr 19 '19 at 16:58
• okay, yes, this is just an obvious reformulation of your description. I haven't seen these permutations studied specifically. It's an interesting question. Apr 19 '19 at 16:59
• See oeis.org/A002464. Apr 21 '19 at 1:03