I'm interested in the number of inequivalent matrices with $\pm 1$ entries, where equivalence means row permutation, column permutation, row multiplication with $-1$, and column multiplication with $-1$?
If $\sim$ is the equivalence relation on $n \times m$ binary matrices such that $A \sim B$ iff one can obtain $B$ by applying any combination of the above operations (row/column permutation, row/column multiplication with $-1$) to $A$, I'm interested in the number of $\sim$-equivalence classes over all $n \times m$ matrices with $\pm 1$ entries. Thank you.