A181119 Number of transpose-complementary plane partitions of n.

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

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

Table of n, a(n) for n=0..10.

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