A181119 Number of transpose-complementary plane partitions of n.

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

Offset: 0

Author

Arvind Ayyer, Jan 21 2011

Keywords

• nonn
• nice

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.

Examples

			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]
		

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

Crossrefs

Cf. A008793, A051255, A078920, A123352.

Programs

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 *)

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

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