A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

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

Offset: 0

Author

Gus Wiseman, Sep 24 2023

Keywords

• nonn
• tabl

Comments

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Examples

			The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   4  0  2  0  1  0
   5  1  0  3  1  1  0
   8  0  3  0  3  0  1  0
  10  2  1  2  2  3  1  1  0
  16  0  5  0  3  0  5  0  1  0
  20  2  2  4  2  6  0  4  1  1  0
  31  0  6  0  8  0  5  0  5  0  1  0
  39  4  4  4  1  6  6  3  2  6  1  1  0
  55  0 13  0  8  0 12  0  6  0  6  0  1  0
  71  5  8  7  3  5  3 16  3  6  0  6  1  1  0
Row n = 6 counts the following partitions:
  (321)     (411)  .  (51)   (33)  (6)  .
  (3111)              (42)
  (2211)              (222)
  (21111)
  (111111)
		

Crossrefs

Row sums are A000041.

The rank statistic counted by this triangle is A325799.

The strict case is A365545, weighted row sums A365922.

The complement (positive subset-sum) is A365658.

Weighted row sums are A365918, for positive subset-sums A304792.

A046663 counts partitions w/o a submultiset summing to k, strict A365663.

A126796 counts complete partitions, ranks A325781, strict A188431.

A364350 counts combination-free strict partitions, complement A364839.

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

A365924 counts incomplete partitions, ranks A365830, strict A365831.

Cf. A000009, A006827, A122768, A364272, A365919, A365921.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n,0,10}, {k,0,n}]