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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.
9

%I #38 Sep 14 2018 16:50:57

%S 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 143164,464836,46592,132,1,7,9724,3636776,75965484,48767805,835584,

%U 429,1,8,43043,91442364,12753712037,55824699632,5900575762,15876096,1430

%N 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 Alois P. Heinz, <a href="/A214722/b214722.txt">Antidiagonals n = 0..20, flattened</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, 1, ...

%e 1, 2, 3, 4, 5, 6, ...

%e 2, 16, 91, 456, 2145, 9724, ...

%e 5, 192, 5471, 143164, 3636776, 91442364, ...

%e 14, 2816, 464836, 75965484, 12753712037, 2214110119572, ...

%e 42, 46592, 48767805, 55824699632, 70692556053053, 98002078234748974, ...

%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], [n]]):

%p seq(seq(A(n, 1+d-n), n=0..d), d=0..10);

%t 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 Columns k=1-4 give: A000108, A006335, A213978, A215220.

%Y Rows n=0-3 give: A000012, A000027, A214824, A211505.

%Y A(n,n) gives A258583.

%Y Cf. A213932, A214637, A214631, A258586.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Jul 26 2012