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A342499 Number of integer partitions of n with strictly decreasing first quotients.

%I A342499 #10 Feb 16 2025 08:34:01
%S A342499 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 A342499 70,78,83,91,102,111,122,133,145,158,170,182,202,217,231,248,268,285,
%U A342499 307,332,354,374,404,436,468,502,537,576,618,654,694,737,782,830
%N A342499 Number of integer partitions of n with strictly decreasing first quotients.
%C A342499 Also the number of reversed partitions of n with strictly decreasing first quotients.
%C A342499 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 A342499 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342499 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342499 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342499 The partition (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
%e A342499 The a(1) = 1 through a(9) = 9 partitions:
%e A342499   (1)  (2)   (3)   (4)   (5)    (6)    (7)    (8)    (9)
%e A342499        (11)  (21)  (22)  (32)   (33)   (43)   (44)   (54)
%e A342499                    (31)  (41)   (42)   (52)   (53)   (63)
%e A342499                          (221)  (51)   (61)   (62)   (72)
%e A342499                                 (321)  (331)  (71)   (81)
%e A342499                                               (332)  (432)
%e A342499                                               (431)  (441)
%e A342499                                                      (531)
%e A342499                                                      (3321)
%t A342499 Table[Length[Select[IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342499 The version for differences instead of quotients is A320470.
%Y A342499 The ordered version is A342494.
%Y A342499 The strictly increasing version is A342498.
%Y A342499 The weakly decreasing version is A342513.
%Y A342499 The strict case is A342518.
%Y A342499 The Heinz numbers of these partitions are listed by A342525.
%Y A342499 A000005 counts constant partitions.
%Y A342499 A000009 counts strict partitions.
%Y A342499 A000041 counts partitions.
%Y A342499 A001055 counts factorizations.
%Y A342499 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342499 A074206 counts ordered factorizations.
%Y A342499 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342499 A342096 counts partitions with adjacent x < 2y (strict: A342097).
%Y A342499 A342098 counts partitions with adjacent parts x > 2y.
%Y A342499 Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K A342499 nonn
%O A342499 0,3
%A A342499 _Gus Wiseman_, Mar 17 2021