Weighted File Placements on Singleton Boards

Abstract: 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.