A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

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

Is column k = n - 7 given by A325695?

Examples

			Triangle begins:
  1
  1  0
  0  1  0
  1  0  1  0
  0  1  0  1  0
  0  0  2  0  1  0
  1  0  0  2  0  1  0
  1  0  0  0  3  0  1  0
  0  1  1  0  0  3  0  1  0
  0  0  3  0  0  0  4  0  1  0
  1  0  0  2  2  0  0  4  0  1  0
  1  0  0  0  5  0  0  0  5  0  1  0
  2  0  0  0  0  5  2  0  0  5  0  1  0
  2  0  1  0  0  0  8  0  0  0  6  0  1  0
  1  1  3  0  0  0  0  7  3  0  0  6  0  1  0
  2  0  4  0  1  0  0  0 12  0  0  0  7  0  1  0
  1  1  2  2  3  1  0  0  0 11  3  0  0  7  0  1  0
  2  0  3  0  7  0  1  0  0  0 16  0  0  0  8  0  1  0
  3  0  0  2  6  3  3  1  0  0  0 15  4  0  0  8  0  1  0
Row n = 12: counts the following partitions:
  (6,3,2,1)  .  .  .  .  (9,2,1)  (6,5,1)  .  .  (11,1)  .  (12)  .
  (5,4,2,1)              (8,3,1)  (6,4,2)        (10,2)
                         (7,4,1)                 (9,3)
                         (7,3,2)                 (8,4)
                         (5,4,3)                 (7,5)
		

Crossrefs

Row sums are A000009, non-strict A000041.

The complement (positive subset-sums) is also A365545 with rows reversed.

Weighted row sums are A365922, non-strict A365918.

The non-strict version is A365923, complement A365658, rank stat A325799.

A046663 counts partitions without a subset 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. A006827, A284640, A304792, A364272, A365921.

Programs

Mathematica

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