@MISC{_onthe, author = {}, title = {On the Minimum Number of}, year = {} }

Share

OpenURL

Abstract

A family P = {π1,..., πq} of permutations of [n] = {1,..., n} is completely k-scrambling [Spencer, 1972; Füredi, 1996] if for any distinct k points x1,..., xk ∈ [n], permutations πi’s in P produce all k! possible orders on πi(x1),..., πi(xk). Let N ∗ (n, k) be the minimum size of such a family. This paper focuses on the case k = 3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. 2 log 2 e log 2 n ≤ N ∗ (n, 3) ≤ 2 log 2 n + (1 + o(1)) log 2 log 2 n. We also prove the existence of limn→ ∞ N ∗ (n, 3)/log 2 n = c3. Determining the value c3 and proving the existence of limn→ ∞ N ∗ (n, k)/log 2 n = ck for k ≥ 4 remain open. 1