  In 2006, Briggs and Remmel gave a factorization theorem for $m$-level rook
placements on singleton boards, a special subset of Ferrers boards.
Subsequently, Barrese, Loehr, Remmel, and Sagan defined the $m$-weighted file
placements to give a combinatorial interpretation to the aforementioned
factorization theorem for all Ferrers boards. An unintended consequence of this
definition is that the sum of the $m$-weights of file placements that are not
$m$-level rook placements on a singleton board must be zero. In this paper, we
attempt to illuminate this result by partitioning the set of file placements
that are not $m$-level rook placements. We do so in such a way that it can be
shown constructively that the sum of the $m$-weights on each partition must be
zero using induction.
