A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Author

Gus Wiseman, Sep 16 2023

Keywords

• nonn
• tabl

Comments

Conjecture: Positions of strictly positive rows are given by A048166.

Examples

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71
		

Links

• Robert Price, Table of n, a(n) for n = 1..300

Crossrefs

Row sums are A000041.

Last column n = k is A126796.

Column k = 3 appears to be A137719.

This is the triangle for the rank statistic A299701.

Central column n = 2k is A365660.

A000009 counts subsets summing to n.

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

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

Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]