A385413 -id:A385413 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Aug 03 2012

			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;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
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

Column k=0 gives: A207542.
Diagonal and lower diagonal give: A000012, A005408.
T(2n,n) gives A385413.
Cf. A214753, A215086.

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]) `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);

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 *)

T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k

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cp's OEIS Frontend

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.

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