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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.

%I A215120 #27 Jun 28 2025 19:46:26
%S A215120 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 A215120 69,11,1,3207,3207,2975,1641,507,101,13,1,17589,17589,16825,10503,
%U A215120 3787,869,139,15,1,102627,102627,100007,69077,28205,7487,1369,183,17,1
%N 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.
%H A215120 Alois P. Heinz, <a href="/A215120/b215120.txt">Rows n = 0..20, flattened</a>
%H A215120 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 A215120 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A215120 T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k<n.
%e A215120 Triangle T(n,k) begins:
%e A215120      1;
%e A215120      1,    1;
%e A215120      3,    3,    1;
%e A215120      9,    9,    5,    1;
%e A215120     33,   33,   23,    7,   1;
%e A215120    135,  135,  109,   43,   9,   1;
%e A215120    633,  633,  557,  261,  69,  11,  1;
%e A215120   3207, 3207, 2975, 1641, 507, 101, 13,  1;
%e A215120   ...
%p A215120 b:= proc(n, k, l) option remember; `if`(n=0, 1,
%p A215120        b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),
%p A215120        b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and
%p A215120        (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
%p A215120        b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
%p A215120        j=1..nops(l[i])), i=1..nops(l)))
%p A215120     end:
%p A215120 A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
%p A215120 H:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):
%p A215120 T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:
%p A215120 seq(seq(T(n, k), k=0..n), n=0..10);
%t A215120 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 A215120 A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
%t A215120 H[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t A215120 T[n_, n_] = 1;
%t A215120 T[n_, k_] := T[n, k] = T[n, k + 1] + H[n, k];
%t A215120 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 28 2022, after _Alois P. Heinz_ *)
%Y A215120 Column k=0 gives: A207542.
%Y A215120 Diagonal and lower diagonal give: A000012, A005408.
%Y A215120 T(2n,n) gives A385413.
%Y A215120 Cf. A214753, A215086.
%K A215120 nonn,tabl
%O A215120 0,4
%A A215120 _Alois P. Heinz_, Aug 03 2012