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

2,2

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)).

Links

Table of n, a(n) for n=2..83.

Reddit, What are these "fake" projective planes?

Example

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;

Crossrefs

Sequence in context: A091088 A335915 A249781 * A115627 A128218 A010283

Adjacent sequences: A342038 A342039 A342040 * A342042 A342043 A342044

Keyword

nonn,tabl

Author

Drake Thomas, Feb 26 2021

Status

approved