A combinatorial theory of higher-dimensional permutation arrays

We define a class of hypercubic (shape [n](d)) arrays that in a certain sen se are d-dimensional analogs of permutation matrices with our motivation fr om algebraic geometry. Various characterizations of permutation arrays are proved. an efficient generation algorithm is given, and enumerative questio ns are discussed although not settled. There is a partial order on the perm utation arrays, specializing to the Bruhat order on S-n, when d equals 2, a nd specializing to the lattice of partitions of a d-set when n equals 2. (C ) 2000 Academic Press.