Posets of annular non-crossing partitions of types B and D

We study the set S nc(p, q) of annular non-crossing permutations of type B, and we introduce a corresponding set NC(p, q) of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between S nc(p, q) and NC (p, q) is a poset isomorphism, where the partial order on S nc(p, q) is induced from the hyperoctahedral group Bp+q, while NC (p, q) is partially ordered by reverse refinement. In the case when q = 1, we prove that NC(p, 1) is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between S nc(p, q) and NC (p, q). For q = 1, the poset NC(p, 1) coincides with a poset “NC(p + 1)” constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.