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A342497
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Number of integer partitions of n with weakly increasing first quotients.
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4
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1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 25, 32, 36, 43, 49, 60, 65, 75, 83, 96, 106, 121, 131, 150, 163, 178, 194, 217, 230, 254, 275, 300, 320, 350, 374, 411, 439, 470, 503, 548, 578, 625, 666, 710, 758, 815, 855, 913, 970, 1029, 1085, 1157, 1212, 1288, 1360
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OFFSET
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0,3
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COMMENTS
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Also called log-concave-up partitions.
Also the number of reversed integer partitions of n with weakly increasing 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 y = (6,3,2,1,1) has first quotients (1/2,2/3,1/2,1) so is not counted under a(13). However, the first differences (-3,-1,-1,0) are weakly increasing, so y is counted under A240026(13).
The a(1) = 1 through a(8) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(211) (311) (51) (61) (62)
(1111) (2111) (222) (322) (71)
(11111) (411) (421) (422)
(3111) (511) (521)
(21111) (4111) (611)
(111111) (31111) (2222)
(211111) (4211)
(1111111) (5111)
(41111)
(311111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], LessEqual@@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 A240026.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The Heinz numbers of these partitions are A342523.
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 all 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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