An upper bound on the number of high-dimensional permutations

Nathan Linial, Zur Luria

Research output: Contribution to journalArticlepeer-review

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 {(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 ≠ 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 (Formula presented.) 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 Brégman’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 Brégman’s theorem.

Original languageAmerican English
Pages (from-to)471-486
Number of pages16
JournalCombinatorica
Volume34
Issue number4
DOIs
StatePublished - 1 Aug 2014

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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