A145835 Number of compositions of the n X n grid graph.

1, 12, 1434, 1691690, 19719299768, 2271230282824746, 2584855762327078145444, 29068227444022728740767607050, 3230042572278849047360048508956727420, 3546545075986984198328715750838554116235343894

Offset: 1

Author

Brian Kell, Oct 21 2008

Keywords

• hard
• nonn

Comments

The number of compositions of the n X n grid graph is the number of partitions of an n X n square into regions of orthogonally connected unit squares.

Examples

			For n=2 the a(2)=12 partitions of a 2x2 square are: 1 partition into a single 2x2 region; 4 partitions into a 3-square 'L' shape and an isolated corner; 2 partitions into 2 1x2 bricks; 4 partitions into a 1x2 brick and 2 isolated squares; and 1 partition into 4 isolated squares.
		

Links

• A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumeration, Integers, 1 (2001), #A4.
• J. N. Ridley and M. E. Mays, Compositions of unions of graphs, Fib. Quart., 42 (2004), 222-230.
• Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. - From _N. J. A. Sloane_, Jan 04 2013

Crossrefs

Cf. A110476.

Extensions

Terms for n=7,8,9 from Frank Simon, Oct 25 2011

a(10) from Frank Simon's thesis added by N. J. A. Sloane, Jan 04 2013