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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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%I #24 Apr 28 2022 03:12:28

%S 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,

%T 69,11,1,3207,3207,2975,1641,507,101,13,1,17589,17589,16825,10503,

%U 3787,869,139,15,1,102627,102627,100007,69077,28205,7487,1369,183,17,1

%N 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.

%H Alois P. Heinz, <a href="/A215120/b215120.txt">Rows n = 0..20, flattened</a>

%H S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229v1 [math.CO], 2012.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

%F T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k<n.

%e Triangle T(n,k) begins:

%e : 1;

%e : 1, 1;

%e : 3, 3, 1;

%e : 9, 9, 5, 1;

%e : 33, 33, 23, 7, 1;

%e : 135, 135, 109, 43, 9, 1;

%e : 633, 633, 557, 261, 69, 11, 1;

%e : 3207, 3207, 2975, 1641, 507, 101, 13, 1;

%p b:= proc(n, k, l) option remember; `if`(n=0, 1,

%p b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),

%p b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and

%p (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),

%p b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),

%p j=1..nops(l[i])), i=1..nops(l)))

%p end:

%p A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):

%p H:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):

%p T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t 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]}]];

%t A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];

%t H[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];

%t T[n_, n_] = 1;

%t T[n_, k_] := T[n, k] = T[n, k + 1] + H[n, k];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 28 2022, after _Alois P. Heinz_ *)

%Y Column k=0 gives: A207542.

%Y Diagonal and lower diagonal give: A000012, A005408.

%Y Cf. A214753, A215086.

%K nonn,tabl

%O 0,4

%A _Alois P. Heinz_, Aug 03 2012