Flag weak order on wreath products

A generating set for the wreath product Zr wreath Sn which leads to a nicely behaved weak order is presented. It is shown that the resulting poset has properties analogous to those of the weak order on the symmetric group: it is a self-dual lattice, ranked by the Foata-Han flag inversion number; any two maximal chains are connected via Tits-type pseudo-Coxeter moves; and its intervals have the desired homotopy types. The associated Möbius function and relevant generating functions are computed.