Non-crossing partitions for classical reflection groups

We introduce analogues of the lattice of non-crossing set partitions for the classical reflection groups of types B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain decompositions (self-dual for type B), whose rank-generating functions, zeta polynomials, rank-selected chain numbers have simple closed forms.