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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 $g_i$?

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Regarding Q1: I believe this is proved in "A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations" by Barcucci et al. https://doi.org/10.1007/s11083-005-9021-x. (Note that 213-avoiding and 312-avoiding permutations are in bijection via inversion, a poset automorphism of the Bruhat order.)

Regarding Q2: I doubt this coincidence will extend in a straightforward way to other types. One reason is because in other types the number of order ideals of the root poset (the "W-Catalan number") is not the same as the number of fully commutative elements of the Weyl group. In Type A, fully commutative elements are 321-avoiding: not exactly the same as 213-avoiding, but equinumerous with them.

EDIT: By the way, even to define what pattern avoidance means in other types is a nontrivial task, but there is such a notion, due to Billey and Postnikov https://doi.org/10.1016/j.aam.2004.08.003.

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    $\begingroup$ Another related paper towards Q1 is Noncrossing partitions and Bruhat order by Thomas Gobet, Nathan Williams. $\endgroup$ Jan 25 at 14:49
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    $\begingroup$ At some point, I (and probably also others) have tried to identify subsets (such as noncrossing partitions or sortable elements) in other Weyl groups W counted by the W-Catalan numbers that yield, under (strong) Bruhat order, the distributive lattice of the root poset. I do not remember any details, but certainly did not succeed in finding such a subset of group elements. $\endgroup$ Jan 25 at 14:51

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