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A342513
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Number of integer partitions of n with weakly decreasing first quotients.
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4
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1, 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 15, 20, 21, 24, 28, 29, 33, 40, 44, 49, 57, 61, 65, 77, 84, 87, 99, 106, 115, 132, 141, 152, 167, 180, 193, 212, 228, 246, 274, 290, 309, 338, 357, 382, 412, 439, 463, 498, 536, 569, 608, 648, 693, 743, 790, 839, 903, 949
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OFFSET
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1,3
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COMMENTS
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Also called log-concave-down partitions.
Also the number of reversed integer partitions of n with weakly decreasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
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LINKS
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EXAMPLE
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The partition (9,7,4,2,1) has first quotients (7/9,4/7,1/2,1/2) so is counted under a(23).
The a(1) = 1 through a(8) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(321) (421) (332)
(111111) (2221) (431)
(1111111) (2222)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], GreaterEqual@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 30}]
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CROSSREFS
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The version for differences instead of quotients is A320466.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The Heinz numbers of these partitions are A342526.
A000005 counts constant partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with adjacent parts x <= 2y.
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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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