[Math] How many $n\times m$ binary matrices are there, up to row and column permutations

Question

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations.

If $\sim$ is the equivalence relation on $n\times m$ binary matrices such that $A \sim B$ iff one can obtain B from applying a permutation matrix to A, I'm interested in the number of $\sim$-equivalence classes over all $n\times m$ binary matrices.

I know there are $2^{nm}$ binary matrices of size $n\times m$, and $n!m!$ possible permutations, but somehow I fail to get an intuition on what this implies for the equivalence classes.

Answer 1

This is solved here using Pólya enumeration theory. For the square case ($n=m$), see this sequence.

Comment: I found these by searching for $1,2,7$ in the OEIS.