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A215120 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. 2
1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 33, 33, 23, 7, 1, 135, 135, 109, 43, 9, 1, 633, 633, 557, 261, 69, 11, 1, 3207, 3207, 2975, 1641, 507, 101, 13, 1, 17589, 17589, 16825, 10503, 3787, 869, 139, 15, 1, 102627, 102627, 100007, 69077, 28205, 7487, 1369, 183, 17, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012.
Wikipedia, Young tableau
FORMULA
T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k<n.
EXAMPLE
Triangle T(n,k) begins:
: 1;
: 1, 1;
: 3, 3, 1;
: 9, 9, 5, 1;
: 33, 33, 23, 7, 1;
: 135, 135, 109, 43, 9, 1;
: 633, 633, 557, 261, 69, 11, 1;
: 3207, 3207, 2975, 1641, 507, 101, 13, 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), [])):
H:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:
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], {}]];
H[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
T[n_, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k + 1] + H[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)
CROSSREFS
Column k=0 gives: A207542.
Diagonal and lower diagonal give: A000012, A005408.
Sequence in context: A193741 A193824 A108075 * A084145 A122919 A188513
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 03 2012
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)