A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 16, 5, 1, 4, 91, 192, 14, 1, 5, 456, 5471, 2816, 42, 1, 6, 2145, 143164, 464836, 46592, 132, 1, 7, 9724, 3636776, 75965484, 48767805, 835584, 429, 1, 8, 43043, 91442364, 12753712037, 55824699632, 5900575762, 15876096, 1430

Offset: 0

Author

Alois P. Heinz, Jul 26 2012

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
   1,     1,        1,           1,              1,                 1, ...
   1,     2,        3,           4,              5,                 6, ...
   2,    16,       91,         456,           2145,              9724, ...
   5,   192,     5471,      143164,        3636776,          91442364, ...
  14,  2816,   464836,    75965484,    12753712037,     2214110119572, ...
  42, 46592, 48767805, 55824699632, 70692556053053, 98002078234748974, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..20, flattened
• S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
• Wikipedia, Young tableau

Crossrefs

Columns k=1-4 give: A000108, A006335, A213978, A215220.

Rows n=0-3 give: A000012, A000027, A214824, A211505.

A(n,n) gives A258583.

Cf. A213932, A214637, A214631, A258586.

Programs

Maple

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
A:= (n, k)-> b([[n$k], [n]]):
seq(seq(A(n, 1+d-n), n=0..d), d=0..10);

Mathematica

b[l_List] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]] ]; a[n_, k_] := b[{Array[n&, k], {n}}]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)