I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore won't have a determinant. While it might also be interesting to ask what number of NxN (0,1)-matrices are singular or non-singular, I'd like to ignore singular matrices altogether in this question.
To get a better grasp on the problem I wrote a computer program to search for the values given an input N. The output is below:
1x1: 2 possible determinants
2x2: 3 ...
3x3: 5 ...
4x4: 9 ...
5x5: 19 ...
Because the program is simply designed to just a brute force over every possible matrix the computation time grows with respect to $O(2^{N^2})$. Computing 6x6 looks like it is going to take me close to a month and 7x7 is beyond hope without access to a cluster. I don't feel like this limited amount of output is enough to make a solid conjecture.
I have a practical application in mind, but I'd also like to get the bounds to satiate my curiosity.
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