|
%I #74 Feb 21 2024 04:30:59
%S 1,2,2,4,12,4,8,74,74,8,16,456,1434,456,16,32,2810,27780,27780,2810,
%T 32,64,17316,538150,1691690,538150,17316,64,128,106706,10424872,
%U 103015508,103015508,10424872,106706,128,256,657552,201947094,6273056950
%N Table of number of partitions of an m X n rectangle, read by descending antidiagonals.
%C 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.
%H Walter Trump, <a href="/A110476/b110476.txt">Table of n, a(n) for n = 1..220</a> (first 40 terms from Hugo van der Sanden).
%H Brian Kell, <a href="/A110476/a110476.txt">Values for m+n < 16</a> [except (7,7), (7,8) and (8,7)]
%H A. Knopfmacher and M. E. Mays, <a href="http://www.emis.de/journals/INTEGERS/papers/b4/b4.Abstract.html">Graph compositions I: Basic enumeration</a>, Integers, 1 (2001), 1-11. [From _Brian Kell_, Oct 21 2008]
%H Yulka Lipkova, Miso Forisek, Tom Zathurecky, and Davidko Pal, <a href="http://ipsc.ksp.sk/contests/ipsc2007/real/problems/d.php">Delicious cake</a>. [From _Brian Kell_, Oct 21 2008]
%H J. N. Ridley and M. E. Mays, <a href="https://www.fq.math.ca/Papers1/42-3/Ridley-Mays-scanned.pdf">Compositions of unions of graphs</a>, Fib. Quart., 42 (2004), 222-230. [From _Brian Kell_, Oct 21 2008]
%H Frank Simon, <a href="https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-101154">Algebraic Methods for Computing the Reliability of Networks</a>, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. - From _N. J. A. Sloane_, Jan 04 2013
%H F. Simon, P. Tittmann and M. Trinks, <a href="https://doi.org/10.37236/501">Counting Connected Set Partitions of Graphs</a>, Electron. J. Combin., 18(1) (2010), #P14, 12pp.
%F a(m,n) = a(n,m).
%F a(1,n) = 2^(n-1) = a(n,1).
%F a(2,n) = A078469(n) = a(n,2).
%F From _Petros Hadjicostas_, Feb 27 2021: (Start)
%F The following two equations seem to follow from the work of _Brian Kell_ and _Frank Simon_:
%F a(3,n) = A108808(n) = a(n,3).
%F a(4,n) = A221157(n) = a(n,4). (End)
%e Array A(m,n) (with rows m >= 1 and columns n >= 1) begins
%e 1, 2, 4, 8, 16, 32, 64, 128, ...
%e 2, 12, 74, 456, 2810, 17316, 106706, ...
%e 4, 74, 1434, 27780, 538150, 10424872, ...
%e 8, 456, 27780, 1691690, 103015508, ...
%e 16, 2810, 538150, 103015508, ...
%e 32, 17316, 10424872, ...
%e 64, 106706, ...
%e 128, ...
%e ...
%Y Cf. A000041, A000079, A078469, A221157.
%Y Cf. A108808, A145835. - _Brian Kell_, Oct 21 2008
%K nonn,tabl
%O 1,2
%A _Hugo van der Sanden_, Sep 08 2005
%E Corrected by Chuck Carroll (chuck(AT)chuckcarroll.org), Jun 06 2006
%E Name edited by _Michel Marcus_, Jul 02 2020
|
