  We study the class of functions on the set of (generalized) Young diagrams
arising as the number of embeddings of bipartite graphs. We give a criterion
for checking when such a function is a polynomial function on Young diagrams
(in the sense of Kerov and Olshanski) in terms of combinatorial properties of
the corresponding bipartite graphs. Our method involves development of a
differential calculus of functions on the set of generalized Young diagrams.
