The following might be a right-level project for a polymath project.
Keith R. Motes, Jonathan P. Olson, Evan J. Rabeaux, Jonathan P. Dowling, S. Jay Olson, Peter P. Rohde proposed in the paper
An amazing formula for the permanent of the matrix representing a sort of the discrete Fourier transform. The formula was reached at by evaluating the cases $n \le 6$ and was checked symbolically for up to $n\le 16$ or so and numerically much beyond. So it must be true! No proofs is known.
Here is the formula: 
$\hat U ^{(n)}$ is a certain version of the discrete Fourier transform defined as follows: 
(For the ordinary matrix of the discrete Fourier transform people did look a little at the permanent but it's not so beautiful.)
Of course, it would be nice to prove it. I talked about it a summer ago with Ron Adin and Oron Propp (an undergraduate from MIT) and we had a few ideas but they did not work. I popularized the problem a little among experts in enumerative combinatorics but I don't know if people are working on it.