A365832 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 0, 1, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 0, 4, 0, 0, 2, 2, 0, 0, 1, 0, 1, 0, 5, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 5, 0, 0, 2, 5, 0, 0, 0, 0, 2

Offset: 0

Author

Gus Wiseman, Sep 28 2023

Keywords

• nonn
• tabl

Examples

			The partition (7,6,1) has sums 1, 6, 7, 8, 13, 14, so is counted under T(14,6).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  0  1
  0  1  0  1  0
  0  1  0  2  0  0
  0  1  0  2  0  0  1
  0  1  0  3  0  0  0  1
  0  1  0  3  0  0  1  1  0
  0  1  0  4  0  0  0  3  0  0
  0  1  0  4  0  0  2  2  0  0  1
  0  1  0  5  0  0  0  5  0  0  0  1
  0  1  0  5  0  0  2  5  0  0  0  0  2
  0  1  0  6  0  0  0  8  0  0  0  1  0  2
  0  1  0  6  0  0  3  7  0  0  0  0  3  1  1
  0  1  0  7  0  0  0 12  0  0  0  1  0  4  0  2
  0  1  0  7  0  0  3 11  0  0  0  1  3  2  2  1  1
  0  1  0  8  0  0  0 16  0  0  0  1  0  7  0  3  0  2
  0  1  0  8  0  0  4 15  0  0  0  1  3  3  6  2  0  0  3
  0  1  0  9  0  0  0 21  0  0  0  2  0  9  0  7  0  1  0  4
  0  1  0  9  0  0  4 20  0  0  1  0  4  8  5  5  0  0  2  0  5
Row n = 14 counts the following partitions (A..E = 10..14):
  (E)  .  (D1)  .  .  (761)  (B21)  .  .  .  .  (6521)  (8321)  (7421)
          (C2)        (752)  (A31)              (6431)
          (B3)        (743)  (941)              (5432)
          (A4)               (932)
          (95)               (851)
          (86)               (842)
                             (653)
		

Crossrefs

Row sums are A000009.

Rightmost column n = k is A188431, non-strict A126796.

The one-based weighted row sums are A284640.

The corresponding rank statistic is A299701.

The non-strict version is A365658.

Central column n = 2k in the non-strict case is A365660.

Reverse-weighted row-sums are A365922, non-strict A276024.

A000041 counts integer partitions.

A000124 counts distinct sums of subsets of {1..n}.

A365543 counts partitions with a submultiset summing to k, strict A365661.

Cf. A046663, A108917, A122768, A137719, A304792, A364916.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,0,15},{k,0,n}]