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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.
2
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jul 21 2012
STATUS
approved