

A181119


Transposecomplementary plane partitions.


3



1, 2, 84, 81796, 1844536720, 962310111888300, 11608208114358751650000, 3236574482779383546336417240000, 20853456581643133066208521560263633137920, 3104385823530881109001458753652585998600603921849920, 10676554307318599842868990948461304923921623250562199975300214736
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OFFSET

0,2


COMMENTS

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.


LINKS

Table of n, a(n) for n=0..10.
R. P. Stanley, Symmetries of Plane Partitions, J. Comb. Theory Ser. A 43 (1986), 103113.
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
Wikipedia, Plane partition


FORMULA

a(n) = binomial(3n1,n)*Product(i=1..2n2,Product(j=i..2n2,(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 GlaisherKinkelin constant.  Vaclav Kotesovec, Feb 28 2015


EXAMPLE

When n=2, there are two transposecomplementary plane partitions,
[1 1] and [2 1], both of whose transpose and complement is equal to themselves.
[1 1] [1 0]


MATHEMATICA

Table[Binomial[3n1, n]Product[(2n+i+j+1)/(i+j+1), {i, 1, 2n2}, {j, i, 2n2}], {n, 0, 10}] (* Harvey P. Dale, Jan 27 2012 *)


PROG

(PARI) a(n) = binomial(3*n1, n)*prod(i=1, 2*n2, prod(j=i, 2*n2, (2*n+i+j+1)/(i+j+1))); \\ Michel Marcus, Jun 18 2015


CROSSREFS

Cf. A008793, A051255, A078920, A123352.
Sequence in context: A215263 A265591 A157063 * A157315 A244947 A078166
Adjacent sequences: A181116 A181117 A181118 * A181120 A181121 A181122


KEYWORD

nonn,nice


AUTHOR

Arvind Ayyer, Jan 21 2011


STATUS

approved



