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%I #9 Feb 21 2023 12:28:07
%S 1,1,2,2,3,4,5,5,7,9,10,11,14,15,18,20,23,26,31,34,39,42,45,51,58,65,
%T 70,78,83,91,102,111,122,133,145,158,170,182,202,217,231,248,268,285,
%U 307,332,354,374,404,436,468,502,537,576,618,654,694,737,782,830
%N Number of integer partitions of n with strictly decreasing first quotients.
%C Also the number of reversed partitions of n with strictly decreasing first quotients.
%C 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).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e The partition (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
%e The a(1) = 1 through a(9) = 9 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (11) (21) (22) (32) (33) (43) (44) (54)
%e (31) (41) (42) (52) (53) (63)
%e (221) (51) (61) (62) (72)
%e (321) (331) (71) (81)
%e (332) (432)
%e (431) (441)
%e (531)
%e (3321)
%t Table[Length[Select[IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y The version for differences instead of quotients is A320470.
%Y The ordered version is A342494.
%Y The strictly increasing version is A342498.
%Y The weakly decreasing version is A342513.
%Y The strict case is A342518.
%Y The Heinz numbers of these partitions are listed by A342525.
%Y A000005 counts constant partitions.
%Y A000009 counts strict partitions.
%Y A000041 counts partitions.
%Y A001055 counts factorizations.
%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A074206 counts ordered factorizations.
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A342096 counts partitions with adjacent x < 2y (strict: A342097).
%Y A342098 counts partitions with adjacent parts x > 2y.
%Y Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 17 2021
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