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A110476 Table of number of partitions of an m X n rectangle, read by antidiagonals, i.e., with entries in the order (m,n) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ... 4
1, 2, 2, 4, 12, 4, 8, 74, 74, 8, 16, 456, 1434, 456, 16, 32, 2810, 27780, 27780, 2810, 32, 64, 17316, 538150, 1691690, 538150, 17316, 64, 128, 106706, 10424872, 103015508, 103015508, 10424872, 106706, 128, 256, 657552, 201947094, 6273056950 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

We count the partitions of the rectangle into regions of orthogonally connected unit squares. a(2, 2) = 12 comprising one partition of the 2 X 2 region; 4 partitions into a 3-square 'L' shape and an isolated corner; 2 partitions into two 1 X 2 bricks; 4 partitions into a 1 X 2 brick and two isolated squares; and 1 partition into four isolated squares.

LINKS

Table of n, a(n) for n=1..40.

Brian Kell, Values for m+n < 16 [except (7,7), (7,8) and (8,7)]

A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumeration, Integers, 1 (2001), 1-11. [From Brian Kell, Oct 21 2008]

Yulka Lipkova, Miso Forisek, Tom Zathurecky, Davidko Pal, Delicious cake [From Brian Kell, Oct 21 2008]

J. N. Ridley and M. E. Mays, Compositions of unions of graphs, Fib. Quart., 42 (2004), 222-230.  [From Brian Kell, Oct 21 2008]

Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer.  nat.), Fakult├Ąt Mathematik und Naturwissenschaften der Technischen Universit├Ąt Dresden, 2012. - From N. J. A. Sloane, Jan 04 2013

F. Simon, P. Tittmann and M. Trinks, Counting Connected Set Partitions of Graphs, Elec. J. Comb. (18), #P14, (2010), 1-12.

FORMULA

a(m, n) = a(n, m); a(1, n) = 2^(n - 1); a(2, n) = A078469(n)

CROSSREFS

Cf. A078469, A000079, A000041.

Cf. A108808, A145835. - Brian Kell, Oct 21 2008

Sequence in context: A219569 A202795 A256890 * A059343 A285944 A112473

Adjacent sequences:  A110473 A110474 A110475 * A110477 A110478 A110479

KEYWORD

nonn,tabl

AUTHOR

Hugo van der Sanden, Sep 08 2005

EXTENSIONS

Corrected by Chuck Carroll (chuck(AT)chuckcarroll.org), Jun 06 2006

STATUS

approved

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)