Let $X$ be the poset of positive roots of a finite root system of Dynkin type $Q$.

Question 1: In Dynkin type $A_n$, is it true that the poset of order ideals of $X$ is isomorphic to the poset of [2,1,3]-avoiding permutations under strong Bruhat order?

I tested this experimentally in Sage, for example:

```
n=4
S=Permutations(n)
W=[p for p in S if p.avoids( [2,1,3])]
B = Poset([W, lambda x,y: x.bruhat_lequal(y)])
O=LatticePoset(B).join_irreducibles_poset()
display(B)
display(O)
```

Question 2: Is there a generalisation to the other Dynkin types so that the poset of order ideals of $X$ is isomorphic to the poset of $\{g_i \}$-avoiding permutations under the strong Bruhat order of the Weyl group for some group elements $g_i$?