# Order ideals of positive root systems and avoiding group elements in the Weyl group

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 $$g_i$$?