An upper bound on the number of high-dimensional permutations

Abstract

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 {(x 1 , . . . , x i−1 , y, x i+1 , . . . , x d+1 )|n ≥ y ≥ 1} for some index d + 1 ≥ i ≥ 1 and some choice of x j ∈ [n] for all j = 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.

An upper bound on the number of Steiner triple systems

“…These relations suggest that there might be deeper analogies to reveal among Latin squares, STS's, and 1-factorizations. Indeed, we have recently proved an asymptotic upper bound on the number of Latin hypercubes [7], and here we prove analogous statements for STS(n) and F(n).…”

Coloured and Directed Designs

“…High-dimensional permutations (also called Latin Hypercubes) are equivalent to K r r+1 -decompositions of K r r+1 (n). In section 2 we will show how the result of [11] implies an approximate formula for the number of such decompositions, thus confirming a conjecture of Linial and Luria [15]. The method applies in greater generality: as an other illustration we will give an approximate formula for the number of generalised Sudoku squares, via H-decompositions of H(n) for an auxiliary 4-graph H.…”

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