On Flag Vectors, the Dowling Lattice, and Braid Arrangements

We study complex hyperplane arrangements whose intersection lattices, known as the Dowling lattices, are a natural generalization of the partition lattice. We give a combinatorial description of the Dowling lattice via enriched partitions to obtain an explicit EL-labeling and then find a recursion for the flag h-vector in terms of weighted derivations. When the hyperplane arrangements are real they correspond to the braid arrangements An and Bn . By applying a result due to Billera and the authors, we obtain a recursive formula for the cd-index of the lattice of regions of the braid arrangements An and Bn .