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A214637 Number of solid standard Young tableaux of shape [[n,n,n],[n,n],[n]].

%I A214637 #18 Jul 19 2017 19:52:13
%S A214637 1,16,17086,61189172,404233159860,3880365678824980,
%T A214637 47959061464818182058,711513280222442751394224,
%U A214637 12121127323153614807021655742,230127245538294682127207785787376,4767460278053986542112719904243778834,106115342273795146740243750912097789131600
%N A214637 Number of solid standard Young tableaux of shape [[n,n,n],[n,n],[n]].
%H A214637 Alois P. Heinz, <a href="/A214637/b214637.txt">Table of n, a(n) for n = 0..25</a>
%H A214637 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 A214637 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A214637 b:= proc(l) option remember; local m; m:= nops(l);
%p A214637       `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
%p A214637       `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
%p A214637       `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
%p A214637        j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
%p A214637     end:
%p A214637 a:= n-> b([[n, n, n], [n, n], [n]]):
%p A214637 seq(a(n), n=0..10);
%t A214637 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_] := b[{{n, n, n}, {n, n}, {n}}]; Table[a[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Dec 18 2013, translated from Maple *)
%Y A214637 Cf. A213932, A213978, A214638.
%K A214637 nonn
%O A214637 0,2
%A A214637 _Alois P. Heinz_, Jul 23 2012