On the Minimum Number of Completely 3-Scrambling Permutations

Tarui, Jun

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005).

Résumé

A family $\mathcal{P} = \{\pi_1, \ldots , \pi_q\}$ of permutations of $[n]=\{1,\ldots,n\}$ is $\textit{completely}$ $k$-$\textit{scrambling}$ [Spencer, 1972; Füredi, 1996] if for any distinct $k$ points $x_1,\ldots,x_k \in [n]$, permutations $\pi_i$'s in $\mathcal{P}$ produce all $k!$ possible orders on $\pi_i (x_1),\ldots, \pi_i(x_k)$. Let $N^{\ast}(n,k)$ 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. $\frac{2}{ \log _2e} \log_2 n \leq N^{\ast}(n,3) \leq 2\log_2n + (1+o(1)) \log_2 \log _2n$. We also prove the existence of $\lim_{n \to \infty} N^{\ast}(n,3) / \log_2 n = c_3$. Determining the value $c_3$ and proving the existence of $\lim_{n \to \infty} N^{\ast}(n,k) / \log_2 n = c_k$ for $k \geq 4$ remain open.

Détail

DOI : 10.46298/dmtcs.3443