An upper bound on the number of high-dimensional permutations

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 × n × . . . n = [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 {(x1, . . . , xi−1, y, xi+1, . . . , xd+1)|n ≥ y ≥ 1} for some index d + 1 ≥ i ≥ 1 and some choice of xj ∈ [n] for all j 6= 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 ddimensional permutations. Our main result is the following upper bound on their number ( (1 + o(1)) n ed )nd . 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 Brègman’s [1] proof of the Minc conjecture on permanents. More concretely, our approach is very close in spirit to Schrijver’s [11] and Radhakrishnan’s [10] proofs of Brègman’s theorem. ∗Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . Supported by ISF and BSF grants. †Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] .