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A325863
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Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum.
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7
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1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 24, 29, 31, 41, 51, 58, 67, 84, 91, 117, 117
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OFFSET
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0,3
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COMMENTS
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A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).
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LINKS
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EXAMPLE
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The partition (2,1,1,1) has non-singleton submultisets {1,2} and {1,1,1} with the same sum, so (2,1,1,1) is not counted under a(5).
The a(1) = 1 through a(8) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(211) (221) (51) (61) (62)
(1111) (311) (222) (322) (71)
(11111) (321) (331) (332)
(411) (421) (422)
(3111) (511) (431)
(111111) (2221) (521)
(4111) (611)
(1111111) (2222)
(3311)
(5111)
(41111)
(11111111)
The 10 non-knapsack partitions counted under a(12):
(7,6,1)
(7,5,2)
(7,4,3)
(7,5,1,1)
(7,4,2,1)
(7,3,3,1)
(7,3,2,2)
(7,4,1,1,1)
(7,2,2,2,1)
(7,1,1,1,1,1,1,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@Plus@@@Union[Subsets[#, {2, Length[#]}]]&]], {n, 0, 15}]
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CROSSREFS
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Cf. A002033, A055212, A143823, A196723, A276024, A299702, A325856, A325862, A325864, A325865, A325866, A325867, A325877.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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