The computational challenge of enumerating high-dimensional rook walks

We present guesses, based on intensive computer algebra calculations, for recurrence equations of the sequences enumerating rook walks in up to twelve dimensions ending on the main diagonal. Computer proofs can in principle be constructed for all of them. For the moment, however, these computations are feasible only for low dimensions. We pose it as a challenge to develop algorithms which can also certify the correctness of the equations we found for the higher dimensions.