login
A214753
Number T(n,k) of solid standard Young tableaux of n cells and height = k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
14
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, 0, 764, 6322, 6716, 2918, 730, 124, 14, 1, 0, 2620, 30930, 40872, 20718, 6118, 1186, 166, 16, 1, 0, 9496, 158008, 255308, 149826, 50056, 11310, 1796, 214, 18, 1
OFFSET
0,5
LINKS
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
T(n,k) = A215086(n,k) - A215086(n,k-1) for k>0, T(n,0) = A215086(n,0) = A000007(n).
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
Columns k=0-10 give: A000007(n), A000085(n) for n>0, A273582, A273583, A273584, A273585, A273586, A273587, A273588, A273589, A273590.
Diagonal and lower diagonal give: A000012, A005843.
Row sums give: A207542.
T(2n,n) gives A273591.
Cf. A215086.
Sequence in context: A183190 A296129 A276544 * A158454 A049243 A077908
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 02 2012
STATUS
approved