Noncrossing partitions and Bruhat order

We prove that the restriction of Bruhat order to type AA noncrossing partitions for the Coxeter element c=s1s2⋯snc=s1s2⋯sn defines a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the base change from the graphical basis of the Temperley–Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we introduce a new set of vectors counted by the Catalan numbers and give new bijections–fixing each reflection–between noncrossing partitions associated to distinct Coxeter elements.