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%I #11 Mar 22 2021 15:01:09
%S 1,1,1,2,2,3,3,5,5,6,7,8,8,11,12,14,15,17,17,21,22,26,29,31,32,35,38,
%T 42,45,48,51,58,59,63,70,76,80,88,94,98,105,113,121,129,133,143,153,
%U 159,166,183,189,195,210,221,231,248,262,273,284,298,312
%N Number of strict integer partitions of n with weakly increasing first quotients.
%C Also called log-concave-up strict partitions.
%C Also the number of reversed strict integer partitions of n with weakly 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 (6,3,2,1) has first quotients (1/2,2/3,1/2) so is not counted under a(12), even though the first differences (-3,-1,-1) are weakly increasing.
%e The a(1) = 1 through a(13) = 11 partitions (A..D = 10..13):
%e 1 2 3 4 5 6 7 8 9 A B C D
%e 21 31 32 42 43 53 54 64 65 75 76
%e 41 51 52 62 63 73 74 84 85
%e 61 71 72 82 83 93 94
%e 421 521 81 91 92 A2 A3
%e 621 532 A1 B1 B2
%e 721 632 732 C1
%e 821 921 643
%e 832
%e 931
%e A21
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y The version for differences instead of quotients is A179255.
%Y The non-strict ordered version is A342492.
%Y The non-strict version is A342497 (ranking: A342523).
%Y The strictly increasing version is A342517.
%Y The weakly decreasing version is A342519.
%Y A000041 counts partitions (strict: A000009).
%Y A000929 counts partitions with all adjacent parts x >= 2y.
%Y A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
%Y Cf. A000005, A003114, A003242, A005117, A057567, A067824, A238710, A253249, A318991, A318992, A342528.
%K nonn
%O 0,4
%A _Gus Wiseman_, Mar 20 2021
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