A342499 - OEIS

%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