I present some constructions on the class of no-broken-circuit (NBC) sets of some "deformations" of the braid arrangement. (The hyperplanes of the deformations have equation of the form xi - xj = k.) This leads us to new bijective proofs for the numbers of regions of the Shi, Linial and other hyperplane arrangements.
This is joint work with Sylvie Corteel and Véronique Ventos.