A384764 Number of uniquely solveable n X m nonograms (hanjie), 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, 384, 210, 32, 1, 1, 64, 816, 3152, 3152, 816, 64, 1, 1, 128, 3206, 24230, 52362, 24230, 3206, 128, 1, 1, 256, 12536, 189898, 814632, 814632, 189898, 12536, 256, 1, 1, 512, 48962, 1473674, 12819322, 25309575, 12819322, 1473674, 48962, 512, 1

Offset: 0

Author

Bertram Felgenhauer, Jun 09 2025

Keywords

• nonn
• tabl
• hard

Comments

In this game there is an n X m grid where each square may or may not be filled. Each column and each row is labeled by the length of each successive block of filled squares, but without indication of the number of unfilled squares in between. The object is to determine which squares are filled.

Examples

			A(2,2) = 16-2 because out of the possible 2^(2*2) grids, only 10/01 and 01/10 have the same row and column clues.
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,    384,     3152,     24230,      189898, ...
  1, 16,  210,   3152,    52362,    814632,    12819322, ...
  1, 32,  816,  24230,   814632,  25309575,   794378773, ...
  1, 64, 3206, 189898, 12819322, 794378773, 49745060669, ...
		

Links

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

Crossrefs

Cf. A242876 (main diagonal), A000012 (column m=0), A000079 (column m=1), A383345 (column m=2).

Cf. A385862 (variant: uniquely solveable n X m yesnograms).

Formula

Basic properties include A(n,m) = A(m,n), A(n,m) <= 2^(n*m), A(0,n) = A(n,0) = 1, and A(1,n) = A(n,1) = 2^n.