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

0,5

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

Formula

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

Example

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;

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])<nops(l[i-1]),
       b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and
       (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
       b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
       j=1..nops(l[i])), i=1..nops(l)))
    end:
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)):
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 *)

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.

Sequence in context: A183190 A296129 A276544 * A158454 A049243 A077908

Adjacent sequences: A214750 A214751 A214752 * A214754 A214755 A214756

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 02 2012

Status

approved