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A338914
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Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.
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24
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1, 0, 0, 1, 1, 2, 3, 4, 6, 9, 11, 16, 23, 29, 39, 53, 69, 90, 118, 150, 195, 249, 315, 398, 506, 629, 789, 982, 1219, 1504, 1860, 2277, 2798, 3413, 4161, 5051, 6137, 7406, 8948, 10765, 12943, 15503, 18571, 22153, 26432, 31432, 37352, 44268, 52444, 61944, 73141
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OFFSET
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0,6
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COMMENTS
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These are also integer partitions that can be partitioned into not necessarily distinct edges (pairs of distinct parts). For example, (3,3,2,2) can be partitioned as {{2,3},{2,3}}, so is counted under a(10), but (4,2,2,2) and (4,2,1,1,1,1) cannot be partitioned into edges. The multiplicities of such a partition form a multigraphical partition (A209816, A320924).
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LINKS
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FORMULA
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EXAMPLE
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The a(3) = 1 through a(10) = 11 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(41) (51) (52) (62) (63) (73)
(2211) (61) (71) (72) (82)
(3211) (3221) (81) (91)
(3311) (3321) (3322)
(4211) (4221) (4321)
(4311) (4411)
(5211) (5221)
(222111) (5311)
(6211)
(322111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]], {n, 0, 30}]
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CROSSREFS
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A096373 counts the complement in even-length partitions.
A320911 gives the Heinz numbers of these partitions.
A339562 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A002100 counts partitions into squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
The following count partitions of even length and give their Heinz numbers:
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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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