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A181119 Transpose-complementary plane partitions 1
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*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.

REFERENCES

Richard Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103-113.

LINKS

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

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)))

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 *)

CROSSREFS

Cf. A008793, A051255, A078920, A123352

Sequence in context: A205643 A215263 A157063 * A157315 A244947 A078166

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 November 24 16:10 EST 2014. Contains 249899 sequences.