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A361907
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Number of integer partitions of n such that (length) * (maximum) > 2*n.
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10
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0, 0, 0, 0, 0, 0, 3, 4, 7, 11, 19, 26, 43, 60, 80, 115, 171, 201, 297, 374, 485, 656, 853, 1064, 1343, 1758, 2218, 2673, 3477, 4218, 5423, 6523, 7962, 10017, 12104, 14409, 17978, 22031, 26318, 31453, 38176, 45442, 55137, 65775, 77451, 92533, 111485, 131057
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OFFSET
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1,7
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COMMENTS
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Also partitions such that (maximum) > 2*(mean).
These are partitions whose complement (see example) has size > n.
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LINKS
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EXAMPLE
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The a(7) = 3 through a(10) = 11 partitions:
(511) (611) (711) (721)
(4111) (5111) (5211) (811)
(31111) (41111) (6111) (6211)
(311111) (42111) (7111)
(51111) (52111)
(411111) (61111)
(3111111) (421111)
(511111)
(3211111)
(4111111)
(31111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not > 2*7, so y is not counted under a(7).
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 is not > 2*8, so y is not counted under a(8).
The partition y = (5,1,1,1) has length 4 and maximum 5, and 4*5 > 2*8, so y is counted under a(8).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 > 2*9, so y is counted under a(9).
The partition y = (3,2,1,1) has diagram:
o o o
o o .
o . .
o . .
with complement (shown in dots) of size 5, and 5 is not > 7, so y is not counted under a(7).
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[#]*Max@@#>2n&]], {n, 30}]
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CROSSREFS
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Reversing the inequality gives A361852.
A051293 counts subsets with integer mean.
A116608 counts partitions by number of distinct parts.
Cf. A027193, A111907, A237752, A237755, A237821, A237824, A237984, A324562, A326622, A327482, A349156, A360071, A361394.
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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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