  A graph is a split graph if its vertex set can be partitioned into a clique
and a stable set. A split graph is unbalanced if there exist two such
partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the
following interesting counting fact: unlabeled, unbalanced split graphs on $n$
vertices can be placed into a bijection with all unlabeled split graphs on
$n-1$ or fewer vertices. In this paper we translate these concepts and the
theorem to different combinatorial settings: minimal set covers, bipartite
graphs with a distinguished block and posets of height one.
