A342041 Triangle read by rows: T(n,k) = maximum number of lines of size k on n points so that every two lines intersect in one point.

1, 3, 1, 3, 1, 1, 4, 2, 1, 1, 5, 4, 1, 1, 1, 6, 7, 2, 1, 1, 1, 7, 7, 2, 1, 1, 1, 1, 8, 7, 3, 2, 1, 1, 1, 1, 9, 7, 5, 2, 1, 1, 1, 1, 1, 10, 7, 6, 2, 2, 1, 1, 1, 1, 1, 11, 7, 9, 3, 2, 1, 1, 1, 1, 1, 1, 12, 7, 13, 3, 2, 2, 1, 1, 1, 1, 1, 1, 13, 7, 13, 4

Offset: 2

Author

Drake Thomas, Feb 26 2021

Keywords

• nonn
• tabl

Comments

Rows start at n = 2, and terms range from k = 2 to k = n. (When k = 1, there can be arbitrarily many lines.)

If a projective plane of order k-1 exists, then for n between k^2-k+1 and k^3-2k^2+3k-2 inclusive, T(n,k) = k^2-k+1. For higher n, T(n,k) = floor((n-1)/(k-1)).

Examples

			For n = 10, k = 4, the unique arrangement with 5 lines (up to symmetry) is
  1111000000
  1000111000
  0100100110
  0010010101
  0001001011
There are no such arrangements with 6 lines. Thus T(10,4) = 5.
These lines are in bijection with the sets of 4 polar axes on a dodecahedron whose endpoints form a cube.
Table begins:
n\k | 2  3  4  5  6  7  8  9
----+-----------------------
  2 | 1;
  3 | 3, 1;
  4 | 3, 1, 1;
  5 | 4, 2, 1, 1;
  6 | 5, 4, 1, 1, 1;
  7 | 6, 7, 2, 1, 1, 1;
  8 | 7, 7, 2, 1, 1, 1, 1;
  9 | 8, 7, 3, 2, 1, 1, 1, 1;
		

Links

• Reddit, What are these "fake" projective planes?