Consider the set of the $N \times M$ binary matrices containing $U$ ones (and $NM-U$ zeros).
How many of them contain at least one $n \times m$ sub-matrix with all ones in it ?
I know pretty well how to compute that in the case of a binary string , i.e. for $N=1$, but could not find yet a suitable strategy to attack the problem in 2D.
Any hint and reference to papers on the subject is welcome.
Your matrices can be considered as adjacency matrices of bipartite graphs : an element $i$ of the set of rows is connected to an element $j$ of the set of columns iff $A_{ij}=1$.
The issue is therefore to enumerate a subset of complete subgraphs therein, called bicliques. It looks to be an active research area with keywords :
"Algorithmic graph theory", "Maximal complete bipartite subgraph", "Biclique", "Consensus"...
See for example
This recent mathSE question (related to Burnside formula) Counting directed bicliques using Burnside's lemma
(https://link.springer.com/article/10.1007/s41109-019-0123-6) where I found for example this early reference : Malgrange Y: Recherche des sous-matrices premières d’une matrice à coefficients binaires. Applications à certains problèmes de graphe. Proceedings of the Deuxième Congrès de l’AFCALTI, pages 231-242 Paris: Gauthier-Villars, 1962
