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A367398
Number of integer partitions of n whose length is not a semi-sum of the parts.
8
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
0,4
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n, 0, 10}]
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
-----------------------------------------------------------
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.
Sequence in context: A279083 A298805 A085147 * A298810 A298811 A298812
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2023
STATUS
approved