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

Author

Alois P. Heinz, Jul 21 2012

Keywords

• nonn
• tabf

Examples

			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, ...
  ...
		

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.

Crossrefs

Row sums give: A005157.

Cf. A000292.

Programs

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);

Formula

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