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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.

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