Symmetric decompositions and the strong Sperner property for noncrossing partition lattices

We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for \(d,n\ge 2\) admit symmetric decompositions into Boolean subposets. As a result, these lattices have the strong Sperner property and their rank-generating polynomials are symmetric, unimodal, and \(\gamma \)-nonnegative. We use computer computations to complete the proof that every noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus answering affirmatively a question raised by D. Armstrong.