A258586 -id:A258586 - cp's OEIS frontend

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

Alois P. Heinz, Jul 26 2012

			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, ...

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

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.

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);

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 *)

A258583 Number of solid standard Young tableaux of shape [[{n}^n],[n]].

Original entry on oeis.org

1, 1, 16, 5471, 75965484, 70692556053053, 6614511157454872712100, 87353366195666890516586545068535, 217757982462900115559339884671224174403391534, 132100470099008733697710444705793312015509514686031193798241, 24431421930145927713526351934816506384811205836191828228900485331569907542066

Offset: 0

Alois P. Heinz, Nov 06 2015

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012.
Wikipedia, Young tableau

Cf. A214722, A258586.

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-> b([[n$n], [n]]):
seq(a(n), n=0..8);

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]]] If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[ l[[i]], j -> l[[i, j]]-1]]], 0], {j, Length[l[[i]]]}], {i, m}]]];
a[n_] := b[{Array[n&, n], {n}}];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Aug 25 2021, after Alois P. Heinz *)

a(n) = A214722(n,n).

cp's OEIS Frontend

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.

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