A214753 Number T(n,k) of solid standard Young tableaux of n cells and height = k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 4, 4, 1, 0, 10, 16, 6, 1, 0, 26, 66, 34, 8, 1, 0, 76, 296, 192, 58, 10, 1, 0, 232, 1334, 1134, 406, 88, 12, 1, 0, 764, 6322, 6716, 2918, 730, 124, 14, 1, 0, 2620, 30930, 40872, 20718, 6118, 1186, 166, 16, 1, 0, 9496, 158008, 255308, 149826, 50056, 11310, 1796, 214, 18, 1

Offset: 0

Author

Alois P. Heinz, Aug 02 2012

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    4,    1;
  0,  10,   16,    6,   1;
  0,  26,   66,   34,   8,  1;
  0,  76,  296,  192,  58, 10,  1;
  0, 232, 1334, 1134, 406, 88, 12,  1;
		

Links

• Alois P. Heinz, Rows n = 0..20, flattened
• S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
• Wikipedia, Young tableau

Crossrefs

Columns k=0-10 give: A000007(n), A000085(n) for n>0, A273582, A273583, A273584, A273585, A273586, A273587, A273588, A273589, A273590.

Diagonal and lower diagonal give: A000012, A005843.

Row sums give: A207542.

T(2n,n) gives A273591.

Cf. A215086.

Programs

Maple

b:= proc(n, k, l) option remember; `if`(n=0, 1,
       b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
T:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n-1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i-1]]], b[n-1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i-1, j]]) && (j == 1 || L[[i, j]] < L[[i, j-1]]), b[n-1, k, ReplacePart[L, i -> ReplacePart[ L[[i]], j -> L[[i, j]]+1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)

Formula

T(n,k) = A215086(n,k) - A215086(n,k-1) for k>0, T(n,0) = A215086(n,0) = A000007(n).