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A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals. 3
1, 1, 1, 1, 2, 1, 1, 6, 16, 1, 1, 20, 936, 192, 1, 1, 70, 85800, 379366, 2816, 1, 1, 252, 9962680, 1825221320, 249664758, 46592, 1, 1, 924, 1340103744, 14336196893200, 89261675900020, 221005209058, 835584, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Antidiagonals n = 0..12

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,    2,         6,             20,                   70, ...

  1,   16,       936,          85800,              9962680, ...

  1,  192,    379366,     1825221320,       14336196893200, ...

  1, 2816, 249664758, 89261675900020, 70351928759681296000, ...

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$(k+1)], [n]$k]):

seq(seq(A(n, d-n), n=0..d), d=0..8);

MATHEMATICA

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_, k_] := b[{Array[n&, k+1], Sequence @@ Array[{n}&, k]}]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-Fran├žois Alcover, Dec 18 2013, translated from Maple *)

CROSSREFS

Columns k=0-2 give: A000012, A006335, A214638.

Rows n=0-1 give: A000012, A000984.

Cf. A213932, A213978, A214637, A214722.

Sequence in context: A094262 A123554 A105291 * A025270 A249450 A331501

Adjacent sequences:  A214628 A214629 A214630 * A214632 A214633 A214634

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 26 2012

STATUS

approved

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Last modified August 6 17:06 EDT 2020. Contains 336255 sequences. (Running on oeis4.)