A342498 - OEIS

%I #10 Feb 21 2023 12:28:31

%S 1,1,2,2,4,4,5,6,8,9,12,12,14,16,18,20,24,26,27,30,35,37,45,47,52,56,

%T 61,65,72,77,83,90,95,99,109,117,127,135,144,151,164,172,181,197,209,

%U 222,239,249,263,280,297,310,332,349,368,391,412,433,457,480,503

%N Number of integer partitions of n with strictly increasing first quotients.

%C Also the number of reversed integer partitions of n with strictly increasing 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 y = (13,7,2,1) has first quotients (7/13,2/7,1/2) so is not counted under a(23). However, the first differences (-6,-5,-1) are strictly increasing, so y is counted under A240027(23).

%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                    (211)  (311)  (51)   (61)   (62)   (72)

%e                                  (411)  (322)  (71)   (81)

%e                                         (511)  (422)  (522)

%e                                                (521)  (621)

%e                                                (611)  (711)

%e                                                       (5211)

%t Table[Length[Select[IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

%Y The version for differences instead of quotients is A240027.

%Y The ordered version is A342493.

%Y The weakly increasing version is A342497.

%Y The strictly decreasing version is A342499.

%Y The strict case is A342517.

%Y The Heinz numbers of these partitions are A342524.

%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