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

Abstract

We study the set S n c 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 n c B ( p , q ) and NC B ( p , q ) is a poset isomorphism, where the partial order on S n c B ( p , q ) is induced from 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 can be pursued in type D, where one gets a canonical isomorphism between S n c D ( p , q ) and NC D ( p , q ) . For q = 1 , the poset NC D ( p , 1 ) coincides with a poset “ N C ( 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.