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A342519 Number of strict integer partitions of n with weakly decreasing first quotients.

%I A342519 #10 Feb 16 2025 08:34:01
%S A342519 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 A342519 66,73,77,89,97,104,115,124,135,147,160,174,193,206,218,238,254,272,
%U A342519 293,313,331,353,381,408,436,468,499,532,569,610,651,694,735,783
%N A342519 Number of strict integer partitions of n with weakly decreasing first quotients.
%C A342519 Also called log-concave-down strict partitions.
%C A342519 Also the number of reversed strict partitions of n with weakly decreasing first quotients.
%C A342519 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 A342519 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342519 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342519 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342519 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 A342519 The a(1) = 1 through a(13) = 14 strict partitions (A..D = 10..13):
%e A342519   1   2   3    4    5    6     7     8     9     A      B     C      D
%e A342519           21   31   32   42    43    53    54    64     65    75     76
%e A342519                     41   51    52    62    63    73     74    84     85
%e A342519                          321   61    71    72    82     83    93     94
%e A342519                                421   431   81    91     92    A2     A3
%e A342519                                            432   541    A1    B1     B2
%e A342519                                            531   631    542   543    C1
%e A342519                                                  4321   641   642    652
%e A342519                                                         731   651    742
%e A342519                                                               741    751
%e A342519                                                               831    841
%e A342519                                                               5421   931
%e A342519                                                                      5431
%e A342519                                                                      6421
%t A342519 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342519 The non-strict ordered version is A069916.
%Y A342519 The version for differences instead of quotients is A320382.
%Y A342519 The non-strict version is A342513 (ranking: A342526).
%Y A342519 The weakly increasing version is A342516.
%Y A342519 The strictly decreasing version is A342518.
%Y A342519 A000005 counts constant partitions.
%Y A342519 A000041 counts partitions (strict: A000009).
%Y A342519 A000929 counts partitions with all adjacent parts x >= 2y.
%Y A342519 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342519 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342519 A057567 counts strict chains of divisors with weakly increasing quotients.
%Y A342519 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342519 A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
%Y A342519 A342528 counts compositions with alternately weakly increasing parts.
%Y A342519 Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.
%K A342519 nonn
%O A342519 0,4
%A A342519 _Gus Wiseman_, Mar 20 2021