A360764 Number T(n,k) of sets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=max(i:T(n,i)>0), read by rows.

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 1, 4, 6, 1, 0, 1, 6, 8, 4, 0, 1, 6, 13, 9, 1, 0, 1, 8, 18, 16, 6, 0, 1, 8, 24, 29, 13, 2, 0, 1, 10, 30, 43, 29, 6, 0, 1, 10, 39, 64, 52, 19, 1, 0, 1, 12, 46, 89, 89, 42, 7, 0, 1, 12, 56, 122, 139, 85, 22, 1

Offset: 0

Author

Alois P. Heinz, Feb 19 2023

Keywords

• nonn
• tabf

Comments

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

Examples

			T(6,1) = 1: {[6]}.
T(6,2) = 4: {[1],[5]}, {[2],[4]}, {[1,5]}, {[2,4]}.
T(6,3) = 6: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[2],[3]}.
T(6,4) = 1: {[1],[2],[1,2]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1,  2;
  0, 1,  2,  1;
  0, 1,  4,  2;
  0, 1,  4,  6,  1;
  0, 1,  6,  8,  4;
  0, 1,  6, 13,  9,  1;
  0, 1,  8, 18, 16,  6;
  0, 1,  8, 24, 29, 13,  2;
  0, 1, 10, 30, 43, 29,  6;
  0, 1, 10, 39, 64, 52, 19, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..250, flattened

Crossrefs

Columns k=0-2 give: A000007, A057427, A052928(n-1) for n>=3.

Row sums give A050342.

Cf. A000009, A008289, A055884, A330462, A360742, A360763.

Programs

Maple

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

Mathematica

h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i<0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]] ;
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)