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A367212
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Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.
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24
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1, 1, 1, 2, 3, 5, 6, 11, 15, 22, 30, 43, 58, 80, 106, 143, 186, 248, 318, 417, 530, 684, 863, 1103, 1379, 1741, 2162, 2707, 3339, 4145, 5081, 6263, 7640, 9357, 11350, 13822, 16692, 20214, 24301, 29300, 35073, 42085, 50208, 59981, 71294, 84866, 100509, 119206
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OFFSET
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0,4
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COMMENTS
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Or, partitions whose length is a subset-sum of the parts.
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LINKS
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EXAMPLE
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The partition (3,2,1,1) has submultisets (3,1) or (2,1,1) with sum 4, so is counted under a(7).
The a(1) = 1 through a(8) = 15 partitions:
(1) (11) (21) (22) (32) (42) (52) (62)
(111) (211) (221) (321) (322) (332)
(1111) (311) (2211) (331) (431)
(2111) (3111) (421) (521)
(11111) (21111) (2221) (2222)
(111111) (3211) (3221)
(4111) (3311)
(22111) (4211)
(31111) (22211)
(211111) (32111)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n, 0, 10}]
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CROSSREFS
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The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
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Triangles:
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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