A369382 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to lattice symmetry.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 1, 3, 0, 2, 0, 3, 0, 6, 0, 10, 0, 6, 0, 9, 0, 12, 1, 18, 2, 9, 0, 5, 0, 7, 0, 8, 0, 12, 0, 18, 0, 14, 0, 17

Offset: 1

Author

Giedrius Alkauskas, Jan 22 2024

Keywords

• nonn
• more

Comments

A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded.

Related to A367783, only sets obtained by rotation and reflection are considered to be the same.

For odd n, a(n) = A367783(n)/8.

For even n, 8 * a(n) >= A367783(n).

a(n) > 0 for even n >= 12.

a(n) > 0 for odd n with natural density 1 (among odd numbers).

Examples

			For n = 4, a(4) = 1 way to place 4 points is as follows:
.xx.
.xx.
For n = 14, a(14) = 1 way to place 14 points is as follows:
  ...x..
  ..x.x.
  .xxx.x
  x.xxx.
  .x.x..
  ..x...
For n = 27, a(27) = 1 way to place 27 points is as follows:
  ....x....
  ...x.....
  ..x......
  .x..xx...
  x..xxxx..
  ..xxxxxxx
  ...xxxxx.
  ....xxx..
  .....x...
		

Links

• Giedrius Alkauskas, Friendly paths for finite subsets of plane integer lattice. I, arXiv:2302.01137 [math.CO], 2024.
• Giedrius Alkauskas, Problem 11484, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182.
• Giedrius Alkauskas, Friendly paths. Problem 11484, Problems and solutions, Amer. Math. Monthly, 119 (2) February (2012), 167-168.

Crossrefs

Cf. A000009, A000292, A005232, A367783.