Bipartite three-uniform hypergraph

A three-uniform hypergraph is bipartite when its vertex set has a partition into two parts such that every hyperedge meets both parts. All such triples on parts of sizes $a,b$ number $a\left(\genfrac{}{}{0ex}{}{b}{2}\right)+b\left(\genfrac{}{}{0ex}{}{a}{2}\right)=ab(a+b-2)/2$. For fixed total order this is maximized by balanced parts.