A215086 Number A(n,k) of solid standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 4, 0, 1, 1, 3, 8, 10, 0, 1, 1, 3, 9, 26, 26, 0, 1, 1, 3, 9, 32, 92, 76, 0, 1, 1, 3, 9, 33, 126, 372, 232, 0, 1, 1, 3, 9, 33, 134, 564, 1566, 764, 0, 1, 1, 3, 9, 33, 135, 622, 2700, 7086, 2620, 0, 1, 1, 3, 9, 33, 135, 632, 3106, 13802, 33550, 9496, 0

Offset: 0

Author

Alois P. Heinz, Aug 02 2012

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1,   1,    1,    1,    1,    1,    1,    1, ...
  0,   1,    1,    1,    1,    1,    1,    1, ...
  0,   2,    3,    3,    3,    3,    3,    3, ...
  0,   4,    8,    9,    9,    9,    9,    9, ...
  0,  10,   26,   32,   33,   33,   33,   33, ...
  0,  26,   92,  126,  134,  135,  135,  135, ...
  0,  76,  372,  564,  622,  632,  633,  633, ...
  0, 232, 1566, 2700, 3106, 3194, 3206, 3207, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..30, 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, A000085, A215087, A320180, A320181, A320182, A320183, A320184, A320185, A320186, A320187.

Rows n=0-1 give: A000012, A057427.

Main diagonal gives A207542.

Cf. A214753.

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), [])):
seq(seq(A(n, d-n), n=0..d), d=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]]] Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]] 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], {}]]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 26 2017, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{i=0..k} A214753(n,i).