A364272 Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 8, 6, 11, 10, 17, 16, 26, 25, 39, 39, 54, 60, 82, 84, 116, 126, 160, 177, 222, 242, 302, 337, 402, 453, 542, 601, 722, 803, 936, 1057, 1234, 1373, 1601, 1793, 2056, 2312, 2658, 2950, 3395, 3789, 4281, 4814, 5452, 6048

Offset: 0

Author

Gus Wiseman, Aug 01 2023

Keywords

• nonn

Comments

First differs from A316402 at a(16) = 11 due to (7,5,3,1).

Examples

			The a(6) = 1 through a(16) = 11 partitions (A=10):
  (321) . (431) . (532)  (5321) (642)  (5431) (743)  (6432)  (853)
                  (541)         (651)  (6421) (752)  (6531)  (862)
                  (4321)        (5421) (7321) (761)  (7431)  (871)
                                (6321)        (5432) (7521)  (6532)
                                              (6431) (9321)  (6541)
                                              (6521) (54321) (7432)
                                              (7421)         (7621)
                                              (8321)         (8431)
                                                             (8521)
                                                             (A321)
                                                             (64321)
		

Crossrefs

The non-strict complement is A237667, ranks A364531.

The non-strict version is A237668, ranks A364532.

The complement in strict partitions is A364349, binary A364533.

The linear combination-free version is A364350.

For subsets of {1..n} we have A364534, complement A151897.

The binary version is A364670, allowing re-used parts A363226.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A108917 counts knapsack partitions, strict A275972, ranks A299702.

A236912 counts binary sum-free partitions, complement A237113.

A323092 counts double-free partitions, ranks A320340.

Cf. A007865, A025065, A085489, A093971, A111133, A240861, A320347, A325862, A363225, A364346.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,30}]