A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 6, 16, 1, 1, 20, 936, 192, 1, 1, 70, 85800, 379366, 2816, 1, 1, 252, 9962680, 1825221320, 249664758, 46592, 1, 1, 924, 1340103744, 14336196893200, 89261675900020, 221005209058, 835584, 1

Offset

0,5

Links

Alois P. Heinz, Antidiagonals n = 0..12

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

Square array A(n,k) begins:
  1,    1,         1,              1,                    1, ...
  1,    2,         6,             20,                   70, ...
  1,   16,       936,          85800,              9962680, ...
  1,  192,    379366,     1825221320,       14336196893200, ...
  1, 2816, 249664758, 89261675900020, 70351928759681296000, ...

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])<j, 0, l[i+1][j]) and l[i][j]>
      `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$(k+1)], [n]$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);

Mathematica

b[l_] := 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+1], Sequence @@ Array[{n}&, k]}]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

Crossrefs

Columns k=0-2 give: A000012, A006335, A214638.

Rows n=0-1 give: A000012, A000984.

Cf. A213932, A213978, A214637, A214722.

Sequence in context: A094262 A123554 A105291 * A025270 A249450 A373343

Adjacent sequences: A214628 A214629 A214630 * A214632 A214633 A214634

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 26 2012

Status

approved