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A290202
Number of solid standard Young tableaux of cylindrical shape lambda X 3, where lambda ranges over all partitions of n.
1
1, 1, 10, 276, 14318, 1214358, 150910592, 25454753376, 5599142988564, 1557618719594808, 532482249378122738, 218108013886160729600, 105215894641522373026220, 59025152558043462549357094, 38095446968224172036448488814, 27985301641485576224718954999962
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 23 2017
STATUS
approved