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%I #33 Sep 14 2018 16:49:38
%S 1,1,1,1,2,1,1,6,16,1,1,20,936,192,1,1,70,85800,379366,2816,1,1,252,
%T 9962680,1825221320,249664758,46592,1,1,924,1340103744,14336196893200,
%U 89261675900020,221005209058,835584,1
%N 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.
%H Alois P. Heinz, <a href="/A214631/b214631.txt">Antidiagonals n = 0..12</a>
%H S. B. Ekhad, D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229v1 [math.CO], 2012
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, ...
%e 1, 2, 6, 20, 70, ...
%e 1, 16, 936, 85800, 9962680, ...
%e 1, 192, 379366, 1825221320, 14336196893200, ...
%e 1, 2816, 249664758, 89261675900020, 70351928759681296000, ...
%p b:= proc(l) option remember; local m; m:= nops(l);
%p `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
%p `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
%p `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
%p j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
%p end:
%p A:= (n, k)-> b([[n$(k+1)], [n]$k]):
%p seq(seq(A(n, d-n), n=0..d), d=0..8);
%t 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 *)
%Y Columns k=0-2 give: A000012, A006335, A214638.
%Y Rows n=0-1 give: A000012, A000984.
%Y Cf. A213932, A213978, A214637, A214722.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Jul 26 2012
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