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A215122 Number T(n,k) of solid standard Young tableaux of shape [[(n-k)*k,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 3
1, 0, 0, 0, 2, 0, 0, 8, 8, 0, 0, 30, 174, 30, 0, 0, 112, 2084, 2084, 112, 0, 0, 420, 21025, 52808, 21025, 420, 0, 0, 1584, 194064, 994788, 994788, 194064, 1584, 0, 0, 6006, 1694224, 16074586, 31497284, 16074586, 1694224, 6006, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
EXAMPLE
Triangle T(n,k) begins:
1;
0, 0;
0, 2, 0;
0, 8, 8, 0;
0, 30, 174, 30, 0;
0, 112, 2084, 2084, 112, 0;
0, 420, 21025, 52808, 21025, 420, 0;
0, 1584, 194064, 994788, 994788, 194064, 1584, 0;
MAPLE
b:= proc(x, y, z) option remember; `if`(z<y, b(x, z, y),
`if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
`if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
end:
T:= (n, k)-> `if`(k=0 xor k=n, 0, b((n-k)*k, k, n-k)):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
b[x_, y_, z_] := b[x, y, z] = If[z<y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; T[n_, k_] := If[k == 0 || k == n, 0, b[(n-k)*k, k, n-k]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
CROSSREFS
Columns k=0-2 give: A000007, A162551(n-1), A215124.
Central elements of rows give A215123.
Sequence in context: A058347 A058547 A230910 * A332473 A235789 A236925
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 03 2012
STATUS
approved

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Last modified February 29 13:01 EST 2024. Contains 370425 sequences. (Running on oeis4.)