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

AbstractWe study the set SncB(p,q) of annular non-crossing permutations of type B, and we introduce a corresponding set NCB(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 SncB(p,q) and NCB(p,q) is a poset isomorphism, where the partial order on SncB(p,q) is induced from the hyperoctahedral group Bp+q, while NCB(p,q) is partially ordered by reverse refinement. In the case when q=1, we prove that NCB(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 SncD(p,q) and NCD(p,q). For q=1, the poset NCD(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 Dn, SIAM Journal of Discrete Mathematics 18 (2004) 397–417], and is a lattice by the results of that paper