A generalization of plexes of Latin squares

A kk-plex of a Latin square is a collection of cells representing each row, column, and symbol precisely kk times. The classic case of k=1k=1 is more commonly known as a transversal  . We introduce the concept of a kk-weight, an integral weight function on the cells of a Latin square whose row, column, and symbol sums are all kk. We then show that several non-existence results about kk-plexes can been seen as more general facts about kk-weights and that the weight analogues of several well-known existence conjectures for plexes actually hold for kk-weights.