A218268 -id:A218268 - cp's OEIS frontend

A182222 Number T(n,k) of standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 10, 10, 9, 4, 1, 26, 26, 25, 16, 5, 1, 76, 76, 75, 56, 25, 6, 1, 232, 232, 231, 197, 105, 36, 7, 1, 764, 764, 763, 694, 441, 176, 49, 8, 1, 2620, 2620, 2619, 2494, 1785, 856, 273, 64, 9, 1, 9496, 9496, 9495, 9244, 7308, 3952, 1506, 400, 81, 10, 1

Offset: 0

Alois P. Heinz, Apr 19 2012

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= k. T(4,3) = 4: 1234, 1243, 1324, 2134; T(3,0) = T(3,1) = 4: 123, 132, 213, 321; T(5,3) = 16: 12345, 12354, 12435, 12543, 13245, 13254, 14325, 14523, 15342, 21345, 21354, 21435, 32145, 34125, 42315, 52341.

			T(4,3) = 4, there are 4 standard Young tableaux of 4 cells and height >= 3:
  +---+   +------+   +------+   +------+
  | 1 |   | 1  2 |   | 1  3 |   | 1  4 |
  | 2 |   | 3 .--+   | 2 .--+   | 2 .--+
  | 3 |   | 4 |      | 4 |      | 3 |
  | 4 |   +---+      +---+      +---+
  +---+
Triangle T(n,k) begins:
    1;
    1,   1;
    2,   2,   1;
    4,   4,   3,   1;
   10,  10,   9,   4,   1;
   26,  26,  25,  16,   5,   1;
   76,  76,  75,  56,  25,   6,  1;
  232, 232, 231, 197, 105,  36,  7,  1;
  764, 764, 763, 694, 441, 176, 49,  8,  1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Columns 0-10 give: A000085, A000085 (for n>0), A001189, A218263, A218264, A218265, A218266, A218267, A218268, A218269, A218262.
Diagonal and lower diagonals give: A000012, A000027(n+1), A000290(n+1) for n>0, A131423(n+1) for n>1.
T(2n,n) gives A318289.
Cf. A047884, A049400, A182172.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
T:= (n, k)-> g(n, n, []) -`if`(k=0, 0, g(n, k-1, [])):
seq(seq(T(n, k), k=0..n), n=0..12);

h[l_] := Module[{n = Length[l]}, Sum[i, {i, l}]! / Product[ Product[1 + l[[i]] - j + Sum [If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
t[n_, k_] := g[n, n, {}] - If[k == 0, 0, g[n, k-1, {}]];
Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

T(n,k) = A182172(n,n) - A182172(n,k-1) for k>0, T(n,0) = A182172(n,n).

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A182222 Number T(n,k) of 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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