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A181119
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Transpose-complementary plane partitions
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1
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1, 2, 84, 81796, 1844536720, 962310111888300, 11608208114358751650000, 3236574482779383546336417240000, 20853456581643133066208521560263633137920, 3104385823530881109001458753652585998600603921849920, 10676554307318599842868990948461304923921623250562199975300214736
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OFFSET
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0,2
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COMMENTS
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The complement of a plane partition inside an m*m*m cube consists of the boxes which are within the cube, but not in the plane partition, rotated in an appropriate way.
a(n) is the number of plane partitions inside an 2n*2n*2n cube whose (matrix) transpose when written as an 2n*2n array is the same as its complement.
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REFERENCES
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Richard Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103-113.
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LINKS
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Table of n, a(n) for n=0..10.
Wikipedia, Plane partition
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FORMULA
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a(n) = binomial(3n-1,n)*Product(i=1..2n-2,Product(j=i..2n-2,(2n+i+j+1)/(i+j+1)))
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EXAMPLE
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When n=2, there are two transpose-complementary plane partitions,
[1 1] and [2 1], both of whose transpose and complement is equal to themselves.
[1 1] [1 0]
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MATHEMATICA
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Table[Binomial[3n-1, n]Product[(2n+i+j+1)/(i+j+1), {i, 1, 2n-2}, {j, i, 2n-2}], {n, 0, 10}] (* From Harvey P. Dale, Jan 27 2012 *)
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CROSSREFS
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Cf. A008793, A051255, A078920, A123352
Sequence in context: A205643 A215263 A157063 * A157315 A078166 A101578
Adjacent sequences: A181116 A181117 A181118 * A181120 A181121 A181122
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KEYWORD
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nonn,nice
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AUTHOR
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Arvind Ayyer, Jan 21 2011
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STATUS
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approved
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