A146971 Number of weight-n binary n X n matrices that yield the all-ones matrix after repeatedly changing a 0 having at least two 1-neighbors to a 1.

1, 2, 14, 130, 1615, 23140, 383820, 7006916, 140537609, 3035127766

Offset: 1

Author

John Tromp, Nov 03 2008

Keywords

• nonn
• more

Comments

There is a proof that the minimum initial weight is n which can be summarized in the single word "perimeter".

Can also be described as the number of percolating sets of size n for 2-neighbor bootstrap percolation in the n X n grid graph; see Balogh, Bollobás and Morris. The large Schröder numbers A006318 count the permutation matrices (one 1 in each row and column) having this property. - Jonathan Noel, Oct 07 2018

Examples

			a(3) = 14 because there are 2, 4, 4 and 4 symmetrical versions of 100 010 001, 100 001 010, 101 000 100 and 101 000 010 respectively.
		

References

• Erik D. Demaine, Martin L. Demaine and Helena A. Verrill, "Coin-Moving Puzzles", in More Games of No Chance, edited by R. J. Nowakowski, 2002, pages 405-431, Cambridge University Press. Collection of papers from the MSRI Combinatorial Game Theory Research Workshop, Berkeley, California, July 24-28, 2000.

Links

• József Balogh, Béla Bollobás and Robert Morris, Bootstrap percolation in high dimensions, arXiv:0907.3097 [math.PR], 2009-2010.
• József Balogh, Béla Bollobás and Robert Morris, Bootstrap percolation in high dimensions, Combin. Probab. Comput. 19 (2010), no. 5-6, 643-692.
• Michael S. Branicky, Python program.
• Erik D. Demaine, Martin L. Demaine and Helena A. Verrill, , PDF version of "Coin-Moving Puzzles"
• Ivars Peterson, Sliding-Coin Puzzles, Science News 163(5), Feb 01, 2003. Description of results in the above paper.

Programs

Python

# see linked program

Extensions

Additional term a(8) from Alvaro Begue's C-program. - John Tromp, Nov 05 2008

Computed a(9) and a(10) with a 128-bitboard based program, the former verifying a result from Alvaro's array based program. - John Tromp, Nov 20 2008