  Goldman, Joichi, and White proved a beautiful theorem showing that the
falling factorial generating function for the rook numbers of a Ferrers board
factors over the integers. Briggs and Remmel studied an analogue of rook
placements where rows are replaced by sets of $m$ rows called levels. They
proved a version of the factorization theorem in that setting, but only for
certain Ferrers boards. We generalize this result to any Ferrers board as well
as giving a p,q-analogue. We also consider a dual situation involving weighted
file placements which permit more than one rook in the same row. In both
settings, we discuss properties of the resulting equivalence classes such as
the number of elements in a class. In addition, we prove analogues of a theorem
of Foata and Sch\"utzenberger giving a distinguished representative in each
class as well as make connections with the q,t-Catalan numbers. We end with
some open questions raised by this work.
