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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]) `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-François 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 December 7 01:11 EST 2023. Contains 367616 sequences. (Running on oeis4.)