A385862 Number of n X m yesnograms that can be solved uniquely, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 52, 52, 16, 1, 1, 32, 210, 368, 210, 32, 1, 1, 64, 816, 2992, 2992, 816, 64, 1, 1, 128, 3206, 23058, 49578, 23058, 3206, 128, 1, 1, 256, 12536, 179576, 775204, 775204, 179576, 12536, 256, 1, 1, 512, 48962, 1388978, 12129616, 24177516, 12129616, 1388978, 48962, 512, 1

Offset: 0

Author

Karl W. Heuer, Aug 06 2025

Keywords

• nonn
• tabl

Comments

In a nonogram puzzle, there is a hidden bitonal grid (or 0/1 matrix), and each row and each column is labeled by the length of each successive block of foreground pixels, but without indication of the number of background pixels separating them; the object is to determine the grid contents. In this variant, called a "yesnogram", the pixel value that represents foreground for row clues is the complement of the value that represents foreground for column clues.

Examples

			For the 3 X 4 grid shown below, the row clues (counting runs of 0s) and the column clues (counting runs of 1s) are sufficient to reconstruct the grid, so this is one of the 2992 solvable grids counted in A(3, 4).
       | 1 3 1 2
   ----+--------
   1   | 0 1 1 1
   1 1 | 0 1 0 1
   2   | 1 1 0 0
Top left corner of the array:
  1,  1,   1,     1,      1,        1,         1, ...
  1,  2,   4,     8,     16,       32,        64, ...
  1,  4,  14,    52,    210,      816,      3206, ...
  1,  8,  52,   368,   2992,    23058,    179576, ...
  1, 16, 210,  2992,  49578,   775204,  12129616, ...
  1, 32, 816, 23058, 775204, 24177516, 754845831, ...
		

Links

• Bertram Felgenhauer, Antidiagonals n+m = 0..13, flattened
• Bertram Felgenhauer, Counting Nonograms.
• Wikipedia, Nonogram.

Crossrefs

Cf. A242876 (solvable n X n nonograms), A384764 (solvable n X m nonograms), A383345 (solvable n X 2 nonograms or yesnograms), A385861 (solvable n X n yesnograms).

Formula

A(0,n) = 1, and A(1,n) = 2^n. A(n,m) = A(m,n), because a grid is solvable iff its complement-transpose is solvable.