OFFSET
0,5
LINKS
Alois P. Heinz, Rows n = 0..20, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
Wikipedia, Young tableau
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 4, 4, 1;
0, 10, 16, 6, 1;
0, 26, 66, 34, 8, 1;
0, 76, 296, 192, 58, 10, 1;
0, 232, 1334, 1134, 406, 88, 12, 1;
MAPLE
b:= proc(n, k, l) option remember; `if`(n=0, 1,
b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),
b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and
(i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
j=1..nops(l[i])), i=1..nops(l)))
end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n-1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i-1]]], b[n-1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i-1, j]]) && (j == 1 || L[[i, j]] < L[[i, j-1]]), b[n-1, k, ReplacePart[L, i -> ReplacePart[ L[[i]], j -> L[[i, j]]+1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 02 2012
STATUS
approved