A182485 Number of partitions of n into exactly k different parts with distinct multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

1, 0, 1, 0, 2, 0, 2, 0, 3, 1, 0, 2, 3, 0, 4, 3, 0, 2, 8, 0, 4, 9, 0, 3, 12, 0, 4, 16, 1, 0, 2, 22, 4, 0, 6, 20, 5, 0, 2, 31, 12, 0, 4, 35, 16, 0, 4, 34, 24, 0, 5, 44, 33, 0, 2, 51, 52, 0, 6, 53, 57, 0, 2, 62, 89, 0, 6, 65, 100, 1, 0, 4, 68, 131, 5, 0, 4, 87

Offset: 0

Author

Alois P. Heinz, May 01 2012

Keywords

• nonn
• tabf

Examples

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,1], [2].
T(4,1) = 3: [1,1,1,1], [2,2], [4].
T(4,2) = 1: [2,1,1]; part 2 occurs once and part 1 occurs twice.
T(5,2) = 3: [2,1,1,1], [2,2,1], [3,1,1].
T(7,2) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1].
T(10,1) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
T(10,3) = 1: [3,2,2,1,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 2;
  0, 2;
  0, 3,  1;
  0, 2,  3;
  0, 4,  3;
  0, 2,  8;
  0, 4,  9;
  0, 3, 12;
  0, 4, 16, 1;
		

Links

• Alois P. Heinz, Rows n = 0..120, flattened

Crossrefs

Columns k=0-7 give: A000007, A000005, A182473, A212114, A212115, A212116, A212117, A212118.

Row sums give: A098859.

First row with length (t+1): A000292(t).

Cf. A242896 (the same for compositions):

Programs

Maple

b:= proc(n, i, t, s) option remember;
      `if`(nops(s)>t, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t, s)+
      add(`if`(j in s, 0, b(n-i*j, i-1, t, s union {j})), j=1..n/i))))
    end:
g:= proc(n) local i; for i while i*(i+1)*(i+2)/6<=n do od; i-1 end:
T:= n-> seq(b(n, n, k, {}) -b(n, n, k-1, {}), k=0..g(n)):
seq(T(n), n=0..30);

Mathematica

b[n_, i_, t_, s_] := b[n, i, t, s] = If[Length[s] > t, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, t, s] + Sum[If[MemberQ[s, j], 0, b[n-i*j, i-1, t, s ~Union~ {j}]], {j, 1, n/i}]]]]; g[n_] := Module[{i}, For[ i = 1, i*(i+1)*(i+2)/6 <= n , i++]; i-1 ]; t[n_] := Table [b[n, n, k, {}] - b[n, n, k-1, {}], {k, 0, g[n]}]; Table [t[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)