A350684 Number T(n,k) of partitions of [n] such that the sum of elements i contained in block i equals k when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=max(0,A008805(n-1)), read by rows.

1, 0, 1, 1, 1, 1, 1, 2, 1, 6, 3, 4, 2, 16, 7, 8, 14, 3, 3, 1, 73, 25, 26, 51, 12, 12, 4, 298, 91, 92, 164, 116, 56, 30, 21, 4, 4, 1, 1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7, 7366, 1797, 1798, 2484, 3228, 1927, 897, 876, 307, 307, 87, 31, 31, 5, 5, 1

Offset: 0

Author

Alois P. Heinz, Jan 11 2022

Keywords

• nonn
• tabf

Examples

			T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
T(4,1) = 3: 432(1), 42(1)|3, 4(1)|3|2.
T(4,2) = 4: 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1,
T(4,3) = 2: 43(1)|(2), 4(1)|3(2).
Triangle T(n,k) begins:
     1;
     0,   1;
     1,   1;
     1,   1,   2,   1;
     6,   3,   4,   2;
    16,   7,   8,  14,   3,   3,   1;
    73,  25,  26,  51,  12,  12,   4;
   298,  91,  92, 164, 116,  56,  30,  21,  4,  4, 1;
  1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-1 give: A350649, A350650.

Row sums give A000110.

Cf. A000217, A008805, A152947, A350647, A350683.

Programs

Maple

b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
     `if`(n=j, x^j, 1)*b(n-1, max(m, j)), j=1..m+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);

Mathematica

b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
     If[n == j, x^j, 1]*b[n - 1, Max[m, j]], {j, 1, m + 1}]]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

Formula

Sum_{k=1..max(0,A008805(n-1))} k * T(n,k) = A350683(n).

T(2n,A000217(n)) = A152947(n+1).

T(2n-1,A000217(n)) = 1 for n>=1.

T(n,2) - T(n,1) = 1 for n>=3.