A342516 - OEIS

%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