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%I #23 Feb 05 2020 01:17:33
%S 1,1,48,6405442434150,213896868423550025518356115338261531036549426
%N Number of solid standard Young tableaux whose shape is a 3-dimensional cube of side length n.
%H Vasilii Duzhin, <a href="/A214953/b214953.txt">Table of n, a(n) for n = 0..5</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>
%p b:= proc(l) option remember; local m; m:= nops(l);
%p `if`({map(x-> x[], l)[]}minus{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-> b([[n$n]$n]):
%p seq(a(n), n=0..4);
%t b[l_] := b[l] = Module[{m = Length[l]}, If[Union[Flatten[l]] ~Complement~ {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}]]];
%t a[0] = 1; a[n_] := b[Table[Table[n, {n}], {n}]];
%t a /@ Range[0, 4] (* _Jean-François Alcover_, Jan 28 2020, after _Alois P. Heinz_ *)
%K nonn,more
%O 0,3
%A _Alois P. Heinz_, Jul 30 2012
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