A290274 Number of solid standard Young tableaux of cylindrical shape lambda X 5, where lambda ranges over all partitions of n.

1, 1, 84, 58604, 118316062, 620383261034, 7137345113624878, 136938419662960675110, 4248619239382421064760418, 208764720295510001353706916224, 15549729565895424021059338656785142, 1588531886834159978895386134546068562294, 215569983507625108792605406075783194767331496

Offset: 0

Author

Alois P. Heinz, Jul 25 2017

Keywords

• nonn

Links

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

Crossrefs

Column k=5 of A215204.

Programs

Maple

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}minus{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:
g:= proc(n, i, l) `if`(n=0 or i=1, b(map(x->[5$x], [l[], 1$n])),
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=0..8);

Mathematica

b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {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}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}] & /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
a[n_] := g[n, n, 5, {}];
Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz in A215204 *)