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A342497 Number of integer partitions of n with weakly increasing first quotients.

%I A342497 #14 Feb 16 2025 08:34:01
%S A342497 1,1,2,3,5,6,9,11,15,18,23,25,32,36,43,49,60,65,75,83,96,106,121,131,
%T A342497 150,163,178,194,217,230,254,275,300,320,350,374,411,439,470,503,548,
%U A342497 578,625,666,710,758,815,855,913,970,1029,1085,1157,1212,1288,1360
%N A342497 Number of integer partitions of n with weakly increasing first quotients.
%C A342497 Also called log-concave-up partitions.
%C A342497 Also the number of reversed integer partitions of n with weakly increasing first quotients.
%C A342497 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 A342497 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342497 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342497 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342497 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).
%e A342497 The a(1) = 1 through a(8) = 15 partitions:
%e A342497   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A342497        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A342497              (111)  (31)    (41)     (42)      (52)       (53)
%e A342497                     (211)   (311)    (51)      (61)       (62)
%e A342497                     (1111)  (2111)   (222)     (322)      (71)
%e A342497                             (11111)  (411)     (421)      (422)
%e A342497                                      (3111)    (511)      (521)
%e A342497                                      (21111)   (4111)     (611)
%e A342497                                      (111111)  (31111)    (2222)
%e A342497                                                (211111)   (4211)
%e A342497                                                (1111111)  (5111)
%e A342497                                                           (41111)
%e A342497                                                           (311111)
%e A342497                                                           (2111111)
%e A342497                                                           (11111111)
%t A342497 Table[Length[Select[IntegerPartitions[n],LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342497 The version for differences instead of quotients is A240026.
%Y A342497 The ordered version is A342492.
%Y A342497 The strictly increasing version is A342498.
%Y A342497 The weakly decreasing version is A342513.
%Y A342497 The strict case is A342516.
%Y A342497 The Heinz numbers of these partitions are A342523.
%Y A342497 A000005 counts constant partitions.
%Y A342497 A000009 counts strict partitions.
%Y A342497 A000041 counts partitions.
%Y A342497 A000929 counts partitions with all adjacent parts x >= 2y.
%Y A342497 A001055 counts factorizations.
%Y A342497 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342497 A074206 counts ordered factorizations.
%Y A342497 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342497 A342094 counts partitions with all adjacent parts x <= 2y.
%Y A342497 Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K A342497 nonn
%O A342497 0,3
%A A342497 _Gus Wiseman_, Mar 17 2021