  We compute the number of ways a given permutation can be written as a product
of exactly $k$ transpositions. We express this number as a linear combination
of explicit geometric sequences, with coefficients which can be computed in
many particular cases. Along the way we prove several symmetry properties for
matrices associated with bipartite graphs, as well as some general (likely
known) properties of Young diagrams. The methods involve linear algebra,
enumeration of border strip tableau, and a differential operator on symmetric
polynomials.
