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 $? 



Enumeration of labeled connected bipartite graphs given partite sets
The number of connected labeled bipartite graphs with bipartition $(X,Y)$ where $|X|=m$ and $|Y|=n$ is the coefficient of $x^my^n/m!\,n!$ in 
$$\log\biggl(\sum\_{m,n=0}^\infty 2^{mn} \frac{x^m}{m!}\frac{y^n}{n!}\biggr).$$
They are sequence [A123260](http://oeis.org/A123260) in the OEIS.


