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

We study the set S(nc)(B)(p, q) of annular non-crossing permutations of type B, and we introduce a corresponding set NC(B) (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)(B)(p, q) and NC(B)(p, q) is a poset isomorphism, where the partial order on S(nc)(B)(p, q) is induced front the hyperoctahedral group B(p+q), while NC(B)(p, q) is partially ordered by reverse refinement. In the case when q = 1, we prove that NC(B)(p, 1) is a lattice with respect to reverse refinement order. We Point out that an analogous development call be pursued in type 1), where one gets a canonical isomorphism between S(nc)(D)(p, q) and NC(D)(p, q). For q = 1, the poset NC(D)(p, 1) coincides with a poset "NC((D))(p + 1)" constructed in a paper by Athanasiadis and Reiner [C.A. Athanasiadis, V. Reiner, Noncrossing partitions for the group D(n), SIAM Journal of Discrete Mathematics 18 (2004) 397-417], and is a lattice by the results of that paper. (C) 2008 Elsevier B.V. All rights reserved.