2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 351-356
| author: | Jun Tarui |
|---|---|
| title: | On the Minimum Number of Completely 3-Scrambling Permutations |
| keywords: | |
| abstract: |
A family
P
= {π
1
,…,π
q
}
[n]={1,…,n}
is completely
k
-scrambling [Spencer, 1972;
Füredi, 1996] if for any distinct
k
points
x
, permutations
1
,…,x
k
∈[n]
π
's in
i
P
k!
possible orders on
π
. Let
i
(x
1
),…,π
i
(x
k
)
N
be the minimum size of such a family. This paper
focuses on the case
*
(n,k)
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 /
We also prove the existence of
log
2
e
log
2
n ≤ N
*
(n,3) ≤2
log
2
n + (1+o(1))
log
2
log
2
n.
lim
n→∞
N
*
(n,3) /
log
2
n = c
3
c
and proving the existence of
3
lim
n→∞
N
*
(n,k) /
log
2
n = c
k
k ≥4
remain open.
|
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| reference: | Jun Tarui (2005), On the Minimum Number of Completely 3-Scrambling Permutations, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 351-356 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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