A290202 Number of solid standard Young tableaux of cylindrical shape lambda X 3, where lambda ranges over all partitions of n.

1, 1, 10, 276, 14318, 1214358, 150910592, 25454753376, 5599142988564, 1557618719594808, 532482249378122738, 218108013886160729600, 105215894641522373026220, 59025152558043462549357094, 38095446968224172036448488814, 27985301641485576224718954999962

Offset

0,3

Links

Table of n, a(n) for n=0..15.

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

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])<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:
g:= proc(n, i, l) `if`(n=0 or i=1, b(map(x->[3$x], [l[], 1$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..10);

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, 3, {}]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz in A215204 *)

Crossrefs

Column k=3 of A215204.

Sequence in context: A368770 A221526 A153331 * A203149 A117654 A067427

Adjacent sequences: A290199 A290200 A290201 * A290203 A290204 A290205

Keyword

nonn

Author

Alois P. Heinz, Jul 23 2017

Status

approved