Enumeration of labeled connected bipartite graphs given partite sets What would be the closed-form expression defining number of all possible labelled connected bipartite graphs given $\mid X \mid = m,  \mid Y \mid = n - m $? 

Let $|X|=x, \ |Y|=y$.
There are $xy$ bipartite graphs, $(x-1)xy$ ways to selected a member of $X$ to be disconnected and form a bipartite graph with the remaining vertices, $(y-1)xy$ with a member of $y$ disconnected, $\frac{1}2xy(y-2)(y-1)$ with two members of $Y$ disconnected etc.
By inclusion-exclusion, the number of labelled connected bipartite graphs is 
$\sum_{i=0}^n (-1)^n \frac{n+1}{n} {x\choose {i+1}}{y\choose{n-i+1}}$, for which there is unlikely to be any simple formula.
