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%I #9 Mar 22 2021 15:01:33
%S 1,1,1,2,2,3,4,5,5,7,8,9,12,14,15,18,18,21,25,29,32,38,40,44,51,57,61,
%T 66,73,77,89,97,104,115,124,135,147,160,174,193,206,218,238,254,272,
%U 293,313,331,353,381,408,436,468,499,532,569,610,651,694,735,783
%N Number of strict integer partitions of n with weakly decreasing first quotients.
%C Also called log-concave-down strict partitions.
%C Also the number of reversed strict partitions of n with weakly 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 strict partition (10,7,4,2,1) has first quotients (7/10,4/7,1/2,1/2) so is counted under a(24), even though the first differences (-3,-3,-2,-1) are weakly increasing.
%e The a(1) = 1 through a(13) = 14 strict 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 321 61 71 72 82 83 93 94
%e 421 431 81 91 92 A2 A3
%e 432 541 A1 B1 B2
%e 531 631 542 543 C1
%e 4321 641 642 652
%e 731 651 742
%e 741 751
%e 831 841
%e 5421 931
%e 5431
%e 6421
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y The non-strict ordered version is A069916.
%Y The version for differences instead of quotients is A320382.
%Y The non-strict version is A342513 (ranking: A342526).
%Y The weakly increasing version is A342516.
%Y The strictly decreasing version is A342518.
%Y A000005 counts constant partitions.
%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 A057567 counts strict chains of divisors with weakly increasing quotients.
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
%Y A342528 counts compositions with alternately weakly increasing parts.
%Y Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.
%K nonn
%O 0,4
%A _Gus Wiseman_, Mar 20 2021
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