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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.

%I A214564 #21 Feb 15 2019 16:19:12
%S A214564 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,
%T A214564 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,
%U A214564 17,15,13,12,10,8,7,5,4,3,2,1,1,1
%N 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.
%H A214564 Alois P. Heinz, <a href="/A214564/b214564.txt">Rows n = 0..21</a>
%H A214564 C. Koutschan, M. Kauers, and D. Zeilberger, <a href="http://www.pnas.org/content/108/6/2196">Proof of George Andrews’s and David Robbins’s q-TSPP conjecture</a>, PNAS (2011), 108: 2196-2199.
%H A214564 R. P. Stanley, <a href="http://dx.doi.org/10.1007/BFb0072521">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
%F A214564 G.f. of row n: Product_{1<=i<=j<=k<=n} (1-q^(i+j+k-1))/(1-q^(i+j+k-2)).
%e A214564 Triangle T(n,k) begins:
%e A214564   1;
%e A214564   1, 1;
%e A214564   1, 1, 1, 1, 1;
%e A214564   1, 1, 1, 2, 2, 2, 2, 2, 1,  1,  1;
%e A214564   1, 1, 1, 2, 3, 3, 4, 5, 5,  5,  6,  5,  5,  5,  4,  3,  3,  2,  1,  1,  1;
%e A214564   1, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 19, 20, 20, 20, 20, 19, ...
%e A214564   ...
%p A214564 gf:= n-> simplify(mul(mul(mul( (1-q^(i+j+k-1))/
%p A214564          (1-q^(i+j+k-2)), i=1..j), j=1..k), k=1..n)):
%p A214564 T:= n-> seq(coeff(gf(n), q, k), k=0..n*(n+1)*(n+2)/6):
%p A214564 seq(T(n), n=0..7);
%Y A214564 Row sums give: A005157.
%Y A214564 Cf. A000292.
%K A214564 nonn,tabf
%O A214564 0,12
%A A214564 _Alois P. Heinz_, Jul 21 2012