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 A181119 Number of transpose-complementary plane partitions of n. 3
 1, 2, 84, 81796, 1844536720, 962310111888300, 11608208114358751650000, 3236574482779383546336417240000, 20853456581643133066208521560263633137920, 3104385823530881109001458753652585998600603921849920, 10676554307318599842868990948461304923921623250562199975300214736 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The complement of a plane partition inside an m X m X 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 X 2n X 2n cube whose (matrix) transpose when written as an 2n X 2n array is the same as its complement. LINKS R. P. Stanley, Symmetries of Plane Partitions, J. Comb. Theory Ser. A 43 (1986), 103-113. P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015. Wikipedia, Plane partition FORMULA a(n) = binomial(3n-1,n)*Product(i=1..2n-2,Product(j=i..2n-2,(2n+i+j+1)/(i+j+1))). a(n) ~ exp(1/24) * 3^(9*n^2 - 3*n/2 - 1/24) / (sqrt(A) * n^(1/24) * 2^(12*n^2 - n - 1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 28 2015 EXAMPLE 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] MATHEMATICA Table[Binomial[3n-1, n]Product[(2n+i+j+1)/(i+j+1), {i, 1, 2n-2}, {j, i, 2n-2}], {n, 0, 10}] (* Harvey P. Dale, Jan 27 2012 *) PROG (PARI) a(n) = binomial(3*n-1, n)*prod(i=1, 2*n-2, prod(j=i, 2*n-2, (2*n+i+j+1)/(i+j+1))); \\ Michel Marcus, Jun 18 2015 CROSSREFS Cf. A008793, A051255, A078920, A123352. Sequence in context: A288312 A289198 A318128 * A293707 A157315 A244947 Adjacent sequences:  A181116 A181117 A181118 * A181120 A181121 A181122 KEYWORD nonn,nice AUTHOR Arvind Ayyer, Jan 21 2011 STATUS approved

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Last modified June 25 02:36 EDT 2021. Contains 345449 sequences. (Running on oeis4.)