Scrambling permutations and entropy of hypergraphs

Abstract

The following result is proved by using entropy of hypergraphs. If r , , . . . , r,, are permutations of the n element set P such that for every triple x , y , z E P, one can find a ri such that ~~( x )is between r i ( y ) and r i ( z ) , then n < exp(d/2). We also study k-scrambling permutations. Several problems remained open. @ 1996 John Wiley & Sons, Inc. MIXING PERMUTATIONS AND A CONTAINMENT PROBLEM OF ORTHANTSThe permutations 7rl, . . . , 7rd of the n-element set P are called 3-mixing if for any 3-element set {i, j , k} C P, one of the permutations places i between j and k, another one puts j between the other two, and the same holds for k, too. Let g(d)denote the maximum n with the above property. This problem is easily seen to be equivalent to the following. What is the largest number n such that one can find an n-point set X C O l d with the property that each orthant orth(x,E) whose origin x belongs to X and whose faces are parallel to the axis contains at most one additional point of X . Here E = (q, e2, . . . , ed) E (1, -l}d and orth(x, E ) is defined as the set of all vectors of the Actually, it is easy to see that g(5) = 7 as was shown by one of the referees as follows. Suppose that there exists an example with five permutations on eight elements. Consider the elements in the positions 1, 2, 7, and 8 of these permutations. There are 20 such places. It is easy to see that no element can occur four times here. So there are at least four elements occurring exactly three times. However, if an element appears three times than all the three must be in positions 2 and 7. However, this is impossible since there are only 10 such places.Let I(n, d ) be the largest number such that for every n-element set P C Rd there exist an x E P and an E E (1, -l } d satisfying lorth(x, E ) n PI 2 1 + I(n, d The proof is given in the next two sections. In Section 4 we show that Iimd-= g(d)L'd exists. Section 5 contains further extremal results, and in Section 6 we propose a series of open problems. THE ENTROPY LEMMALet 9 be a multihypergraph with (the finite) underlying set (or vertex set) V, i.e., it is a collection of subsets of V, 9 = { F l , F2, . . . , Fm}, where repetition of the members is allowed. The multiplicity of the set S C V is denoted by p(S) [or p(S, 9) to be precise], and set p = p ( 9 ) the maximum multiplicity, p = maxScv p(S). Here p(S) is a nonnegative integer with Cscv p(S) = m. Define the entropy function H( y) = y log,( 1 ly) + (1 -y) log,( 1 /( 1 -y)) for all 0 < y < 1, H ( 0 ) = H(1) = 0. Then H is a concave real function. The binary entropy of the hypergraph 9 is defined as We are going to use the following lemma which was first proved (for p = 1) by Kleitman, Shearer, and Sturtevant [ I l l . The proof for multihypergraphs is PROOF OF THEOREM 1.2From now on we consider only the permutation version of the problem. Let P be an n-element set, it is usually identified with [n] := {1,2, . . . , n}. The matrix M = [ M i , j ] is called a d X n permutation matrix if each of its d rows contain...

A group-based structure for perfect sequence covering arrays

“…Although our interest in this paper is in determining the exact value of g(n, k) for small n and k, we summarize in Theorems 2.10 and 2.11 below the best known asymptotic bounds on the growth rate of g(n, k) as n and k grow. These results improve on previous asymptotic results for completely scrambling sets [19,10,6,18]. Theorem 2.10 holds for general k, and was proved by combining combinatorial arguments with a result due to Wilson [22,Thm.…”

Shattering $k$-sets with Permutations

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The covering threshold of a directed acyclic graph by directed acyclic subgraphs

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