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A328960
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Number of integer partitions of n whose number of nontrivial submultisets is greater than their number of distinct parts times their number of parts minus 1.
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6
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0, 0, 0, 0, 0, 0, 1, 2, 6, 10, 18, 28, 45, 63, 93, 129, 178, 238, 321, 419, 551, 708, 911, 1158, 1472, 1845, 2316, 2883, 3583, 4421, 5453, 6680, 8180, 9964, 12122, 14687, 17771, 21418, 25788, 30949, 37092, 44324, 52906, 62980, 74885, 88832, 105243, 124429
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OFFSET
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0,8
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COMMENTS
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These partitions are conjectured to be precisely those that have a pair of multiset partitions such that no part of one is a submultiset of any part of the other (see A320632). For example, such a pair of partitions of {1,1,2,2} is ({{1,1},{2,2}}, {{1,2},{1,2}}).
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LINKS
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EXAMPLE
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The a(6) = 1 through a(10) = 18 partitions:
(2211) (3211) (3221) (3321) (3322)
(22111) (3311) (4221) (4321)
(4211) (4311) (4411)
(22211) (5211) (5221)
(32111) (32211) (5311)
(221111) (33111) (6211)
(42111) (32221)
(222111) (33211)
(321111) (42211)
(2211111) (43111)
(52111)
(222211)
(322111)
(331111)
(421111)
(2221111)
(3211111)
(22111111)
For example, the partition (4,2,2,1,1) has 16 nontrivial submultisets: {(1), (2), (4), (11), (21), ..., (2211), (4211), (4221)}, and 5 parts, 3 of which are distinct. Since 16 > (5 - 1) * 3 = 12, the partition (42211) is counted under a(10)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], 0<Times@@(1+Length/@Split[#])-2-(Length[#]-1)*Length[Union[#]]&]], {n, 0, 30}]
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CROSSREFS
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The Heinz numbers of these partitions are conjectured to be A320632.
A307409(n) is (omega(n) - 1) * nu(n).
A328958(n) is sigma_0(n) - omega(n) * nu(n).
A328959(n) is sigma_0(n) - 2 - (omega(n) - 1) * nu(n).
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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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