... subgroup of S2n,actingon{−n, −n + 1, , 1, 1, 2, ,n},whichcommutes with the sign change i → −i. The reflections are the transpositions (i − i) and the permutations (ij)(−i − j ), with i = j, which ... the set { 1, 2, ,in,jn+ 1, ,n+1},andthecycle(in+1 jn)is the identical permutation. Thus we have(1 2 inin+2 n+1)=(i1j1) (in−1jn−1).Relabeling in+ 2, ,n+1asin+ 1, ,n, we get ... S2nto partitions of {−n, −n + 1, , 1, 1, 2, ,n} defined above restricts to a bijection from NCWntoNCBn , see [G ], where this is used to recover the type B analogue of the main result in[B2]....