  What is the higher-dimensional analog of a permutation? If we think of a
permutation as given by a permutation matrix, then the following definition
suggests itself: A d-dimensional permutation of order n is an [n]^(d+1) array
of zeros and ones in which every "line" contains a unique 1 entry. A line here
is a set of entries of the form {(x_1,...,x_{i-1},y,x_{i+1},...,x_{d+1})}, for
y between 1 and n, some index i between 1 and d+1 and some choice of x_j in [n]
for all j except i. It is easy to observe that a one-dimensional permutation is
simply a permutation matrix and that a two-dimensional permutation is
synonymous with an order-n Latin square. We seek an estimate for the number of
d-dimensional permutations. Our main result is the following upper bound on
their number: ((1+o(1))(n/e^d))^(n^d).
  We tend to believe that this is actually the correct number, but the problem
of proving the complementary lower bound remains open. Our main tool is an
adaptation of Bregman's proof of the Minc conjecture on permanents. More
concretely, our approach is very close in spirit to Radhakrishnan's proof of
Bregman's theorem.
