A076523 Maximal number of halving lines for 2n points in plane.

1, 3, 6, 9, 13, 18, 22, 27, 33, 38, 44, 51, 57

Offset

1,2

Comments

Let S be a set of n points in the plane. A halving line is a line through two points in S that splits the remaining points into two equal-sized subsets. How many halving lines can S have?

The values n = 8, 9, 10, 11, 12 and 13 were obtained by Abrego et al. The same values hold also for the maximum number of pseudo-halving lines in a generalized configuration of 2n points. The next unknown value, n = 14 (i.e. the maximum number of halving lines among 28 points), is either 63 or 64. - Bernardo M Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

Links

Table of n, a(n) for n=1..13.

B. M. Abrego, S. Fernandez-Merchant, J. Leaños and G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27, Electronic Notes in Discrete Mathematics, 30 (2008), 261-266.

A. Beygelzimer and S. Radziszowski, On halving line arrangements, Discrete Math., 257 (2002), 267-283.

Jeff Erickson, Halving lines and k-sets

Tanya Khovanova and Dai Yang, Halving Lines and Their Underlying Graphs, arXiv:1210.4959 [math.CO], 2012.

Tanya Khovanova, Dai Yang, Connected Components of Underlying Graphs of Halving Lines, arXiv:1304.5658 [math.CO], 2013.

Tanya Khovanova, Dai Yang, Fission of Halving Edges Graphs, arXiv:1310.3510 [math.CO], 2013.

Geza Toth, Point sets with many k-sets, in Proceedings of the 16th Annual ACM Symposium on Computational Geometry, 2000, pp. 37-42.

Crossrefs

Sequence in context: A032782 A310157 A310158 * A310159 A282720 A129403

Adjacent sequences: A076520 A076521 A076522 * A076524 A076525 A076526

Keyword

nonn,more

Author

N. J. A. Sloane, Oct 18 2002

Extensions

More terms from Bernardo M Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

Status

approved