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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. 9
1, 1, 1, 1, 2, 2, 1, 3, 16, 5, 1, 4, 91, 192, 14, 1, 5, 456, 5471, 2816, 42, 1, 6, 2145, 143164, 464836, 46592, 132, 1, 7, 9724, 3636776, 75965484, 48767805, 835584, 429, 1, 8, 43043, 91442364, 12753712037, 55824699632, 5900575762, 15876096, 1430 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Antidiagonals n = 0..20, flattened

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012

Wikipedia, Young tableau

EXAMPLE

Square array A(n,k) begins:

   1,     1,        1,           1,              1,                 1, ...

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

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

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

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

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

MAPLE

b:= proc(l) option remember; local m; m:= nops(l);

      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>

      `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>

      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(

       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))

    end:

A:= (n, k)-> b([[n$k], [n]]):

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

MATHEMATICA

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-Fran├žois Alcover, Dec 17 2013, translated from Maple *)

CROSSREFS

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

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

A(n,n) gives A258583.

Cf. A213932, A214637, A214631, A258586.

Sequence in context: A258222 A112324 A061531 * A071430 A092514 A106641

Adjacent sequences:  A214719 A214720 A214721 * A214723 A214724 A214725

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 26 2012

STATUS

approved

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)