A367398 Number of integer partitions of n whose length is not a semi-sum of the parts.

1, 1, 1, 3, 4, 6, 8, 12, 16, 23, 28, 41, 52, 71, 89, 122, 151, 200, 246, 321, 398, 510, 620, 794, 968, 1212, 1474, 1837, 2219, 2748, 3302, 4055, 4882, 5942, 7094, 8623, 10275, 12376, 14721, 17661, 20920, 25011, 29516, 35120, 41419, 49053, 57609, 68092, 79780

Offset: 0

Author

Gus Wiseman, Nov 19 2023

Keywords

• nonn

Comments

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Examples

			For the partition y = (4,3,1) we have semi-sums {4,5,7}, which do not include 3 (the length of y), so y is counted under a(8).
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (21)   (22)    (32)     (33)      (43)       (44)
            (111)  (31)    (41)     (42)      (52)       (53)
                   (1111)  (311)    (51)      (61)       (62)
                           (2111)   (222)     (322)      (71)
                           (11111)  (411)     (331)      (332)
                                    (21111)   (511)      (422)
                                    (111111)  (4111)     (431)
                                              (22111)    (611)
                                              (31111)    (4211)
                                              (211111)   (5111)
                                              (1111111)  (22211)
                                                         (221111)
                                                         (311111)
                                                         (2111111)
                                                         (11111111)
		

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.

sum-full sum-free comb-full comb-free semi-full semi-free

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partitions: A367212 A367213 A367218 A367219 A367394 A367398*

strict: A367214 A367215 A367220 A367221 A367395 A367399

subsets: A367216 A367217 A367222 A367223 A367396 A367400

ranks: A367224 A367225 A367226 A367227 A367397 A367401

A000041 counts partitions, strict A000009.

A002865 counts partitions whose length is a part, complement A229816.

A236912 counts partitions containing no semi-sum, ranks A364461.

A237113 counts partitions containing a semi-sum, ranks A364462.

A237667 counts sum-free partitions, sum-full A237668.

A366738 counts semi-sums of partitions, strict A366741.

A367402 counts partitions with covering semi-sums, complement A367403.

Triangles:

A008284 counts partitions by length, strict A008289.

A365541 counts subsets with a semi-sum k.

A367404 counts partitions with a semi-sum k, strict A367405.

Cf. A000700, A126796, A304792, A363225, A364272, A365543.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#,{2}],Length[#]]&]],{n,0,10}]