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A367402
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Number of integer partitions of n whose semi-sums cover an interval of positive integers.
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5
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1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 26, 31, 38, 44, 58, 64, 81, 95, 116, 137, 166, 192, 233, 278, 330, 385, 459, 542, 636, 759, 879, 1038, 1211, 1418, 1656, 1942, 2242, 2618, 3029, 3535, 4060, 4735, 5429, 6299, 7231, 8346, 9556, 11031, 12593, 14482, 16525
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
The a(1) = 1 through a(8) = 13 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) (2111) (222) (322) (71)
(11111) (321) (2221) (332)
(2211) (3211) (2222)
(21111) (22111) (3221)
(111111) (211111) (22211)
(1111111) (32111)
(221111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#, {2}]; If[d=={}, {}, Range[Min@@d, Max@@d]]==Union[d])&]], {n, 0, 15}]
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CROSSREFS
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The complement is counted by A367403.
A000009 counts partitions covering an initial interval, ranks A055932.
A086971 counts semi-sums of prime indices.
A261036 counts complete partitions by maximum.
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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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