OFFSET
0,3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 23 2017
STATUS
approved