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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.

%I A214722 #38 Sep 14 2018 16:50:57
%S A214722 1,1,1,1,2,2,1,3,16,5,1,4,91,192,14,1,5,456,5471,2816,42,1,6,2145,
%T A214722 143164,464836,46592,132,1,7,9724,3636776,75965484,48767805,835584,
%U A214722 429,1,8,43043,91442364,12753712037,55824699632,5900575762,15876096,1430
%N 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.
%H A214722 Alois P. Heinz, <a href="/A214722/b214722.txt">Antidiagonals n = 0..20, flattened</a>
%H A214722 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 A214722 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A214722 Square array A(n,k) begins:
%e A214722    1,     1,        1,           1,              1,                 1, ...
%e A214722    1,     2,        3,           4,              5,                 6, ...
%e A214722    2,    16,       91,         456,           2145,              9724, ...
%e A214722    5,   192,     5471,      143164,        3636776,          91442364, ...
%e A214722   14,  2816,   464836,    75965484,    12753712037,     2214110119572, ...
%e A214722   42, 46592, 48767805, 55824699632, 70692556053053, 98002078234748974, ...
%p A214722 b:= proc(l) option remember; local m; m:= nops(l);
%p A214722       `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
%p A214722       `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
%p A214722       `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
%p A214722        j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
%p A214722     end:
%p A214722 A:= (n, k)-> b([[n$k], [n]]):
%p A214722 seq(seq(A(n, 1+d-n), n=0..d), d=0..10);
%t A214722 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 *)
%Y A214722 Columns k=1-4 give: A000108, A006335, A213978, A215220.
%Y A214722 Rows n=0-3 give: A000012, A000027, A214824, A211505.
%Y A214722 A(n,n) gives A258583.
%Y A214722 Cf. A213932, A214637, A214631, A258586.
%K A214722 nonn,tabl
%O A214722 0,5
%A A214722 _Alois P. Heinz_, Jul 26 2012