A214564 Number T(n,k) of totally symmetric plane partitions with largest part <= n and exactly k orbits under action of the symmetric group S_3; triangle T(n,k), n>=0, 0<=k<=A000292(n), read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 19, 20, 20, 20, 20, 19, 18, 17, 15, 13, 12, 10, 8, 7, 5, 4, 3, 2, 1, 1, 1

Offset

0,12

Links

Alois P. Heinz, Rows n = 0..21

C. Koutschan, M. Kauers, and D. Zeilberger, Proof of George Andrews’s and David Robbins’s q-TSPP conjecture, PNAS (2011), 108: 2196-2199.

R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

Formula

G.f. of row n: Product_{1<=i<=j<=k<=n} (1-q^(i+j+k-1))/(1-q^(i+j+k-2)).

Example

Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1, 1, 1, 1;
  1, 1, 1, 2, 2, 2, 2, 2, 1,  1,  1;
  1, 1, 1, 2, 3, 3, 4, 5, 5,  5,  6,  5,  5,  5,  4,  3,  3,  2,  1,  1,  1;
  1, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 19, 20, 20, 20, 20, 19, ...
  ...

Maple

gf:= n-> simplify(mul(mul(mul( (1-q^(i+j+k-1))/
         (1-q^(i+j+k-2)), i=1..j), j=1..k), k=1..n)):
T:= n-> seq(coeff(gf(n), q, k), k=0..n*(n+1)*(n+2)/6):
seq(T(n), n=0..7);

Crossrefs

Row sums give: A005157.

Cf. A000292.

Sequence in context: A016016 A345112 A063059 * A102675 A177849 A143544

Adjacent sequences: A214561 A214562 A214563 * A214565 A214566 A214567

Keyword

nonn,tabf

Author

Alois P. Heinz, Jul 21 2012

Status

approved