Zur Luria

An upper bound on the number of high-dimensional permutations

AbstractWhat 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

On the Vertices of the d-Dimensional Birkhoff Polytope

\left( {(1 + o(1))\frac{n} {{e^d }}} \right)^{n^d } .

Discrepancy of High-Dimensional Permutations

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.

Chernoff's Inequality - A very elementary proof

Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n -vertex Steiner triple systems: . Our main result is that The proof is based on the entropy method. As a prelude to this proof we consider the number F(n) of 1 -factorizations of the complete graph on n vertices. Using the Kahn-Lovasz theorem it can be shown that We show how to derive this bound using the entropy method. Both bounds are conjectured to be sharp.

New bounds on the number of n-queens configurations

Let us denote by Ωn the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let

On the Threshold Problem for Latin Boxes

\varOmega^{(2)}_{n}

A Sharp Threshold for Spanning 2-Spheres in Random 2-Complexes

be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entries in which every line sums to 1. What can be said about

On simple connectivity of random 2-complexes.

\varOmega ^{(2)}_{n}