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A110476
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Table of number of partitions of an m X n rectangle, read by descending antidiagonals.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(m,n) = a(n,m).
a(1,n) = 2^(n-1) = a(n,1).
a(4,n) = A221157(n) = a(n,4). (End)
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EXAMPLE
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Array A(m,n) (with rows m >= 1 and columns n >= 1) begins
1, 2, 4, 8, 16, 32, 64, 128, ...
2, 12, 74, 456, 2810, 17316, 106706, ...
4, 74, 1434, 27780, 538150, 10424872, ...
8, 456, 27780, 1691690, 103015508, ...
16, 2810, 538150, 103015508, ...
32, 17316, 10424872, ...
64, 106706, ...
128, ...
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Corrected by Chuck Carroll (chuck(AT)chuckcarroll.org), Jun 06 2006
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STATUS
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approved
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