cp's OEIS Frontend

A342516 Number of strict integer partitions of n with weakly increasing first quotients.

%I A342516 #12 Feb 16 2025 08:34:01
%S A342516 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 A342516 42,45,48,51,58,59,63,70,76,80,88,94,98,105,113,121,129,133,143,153,
%U A342516 159,166,183,189,195,210,221,231,248,262,273,284,298,312
%N A342516 Number of strict integer partitions of n with weakly increasing first quotients.
%C A342516 Also called log-concave-up strict partitions.
%C A342516 Also the number of reversed strict integer partitions of n with weakly increasing first quotients.
%C A342516 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 A342516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342516 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342516 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342516 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 A342516 The a(1) = 1 through a(13) = 11 partitions (A..D = 10..13):
%e A342516   1   2   3    4    5    6    7     8     9     A     B     C     D
%e A342516           21   31   32   42   43    53    54    64    65    75    76
%e A342516                     41   51   52    62    63    73    74    84    85
%e A342516                               61    71    72    82    83    93    94
%e A342516                               421   521   81    91    92    A2    A3
%e A342516                                           621   532   A1    B1    B2
%e A342516                                                 721   632   732   C1
%e A342516                                                       821   921   643
%e A342516                                                                   832
%e A342516                                                                   931
%e A342516                                                                   A21
%t A342516 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342516 The version for differences instead of quotients is A179255.
%Y A342516 The non-strict ordered version is A342492.
%Y A342516 The non-strict version is A342497 (ranking: A342523).
%Y A342516 The strictly increasing version is A342517.
%Y A342516 The weakly decreasing version is A342519.
%Y A342516 A000041 counts partitions (strict: A000009).
%Y A342516 A000929 counts partitions with all adjacent parts x >= 2y.
%Y A342516 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342516 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342516 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342516 A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
%Y A342516 Cf. A000005, A003114, A003242, A005117, A057567, A067824, A238710, A253249, A318991, A318992, A342528.
%K A342516 nonn
%O A342516 0,4
%A A342516 _Gus Wiseman_, Mar 20 2021