cp's OEIS Frontend

A342518 Number of strict integer partitions of n with strictly decreasing first quotients.

%I A342518 #12 Feb 16 2025 08:34:01
%S A342518 1,1,1,2,2,3,4,4,5,7,8,9,11,12,13,17,18,21,24,28,30,34,37,41,47,52,56,
%T A342518 63,68,72,83,89,99,108,117,128,139,149,163,179,189,203,217,233,250,
%U A342518 272,289,305,329,355,381,410,438,471,505,540,571,607,645,683,726
%N A342518 Number of strict integer partitions of n with strictly decreasing first quotients.
%C A342518 Also the number of reversed strict integer partitions of n with strictly decreasing first quotients.
%C A342518 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 A342518 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342518 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342518 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342518 The strict partition (12,10,6,3,1) has first quotients (5/6,3/5,1/2,1/3) so is counted under a(32), even though the differences (-2,-4,-3,-2) are not strictly decreasing.
%e A342518 The a(1) = 1 through a(13) = 12 partitions (A..D = 10..13):
%e A342518   1   2   3    4    5    6     7    8     9     A      B     C     D
%e A342518           21   31   32   42    43   53    54    64     65    75    76
%e A342518                     41   51    52   62    63    73     74    84    85
%e A342518                          321   61   71    72    82     83    93    94
%e A342518                                     431   81    91     92    A2    A3
%e A342518                                           432   541    A1    B1    B2
%e A342518                                           531   631    542   543   C1
%e A342518                                                 4321   641   642   652
%e A342518                                                        731   651   742
%e A342518                                                              741   751
%e A342518                                                              831   841
%e A342518                                                                    5431
%t A342518 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342518 The version for differences instead of quotients is A320388.
%Y A342518 The version for chains of divisors is A342086 (non-strict: A057567).
%Y A342518 The non-strict ordered version is A342494.
%Y A342518 The non-strict version is A342499 (ranking: A342525).
%Y A342518 The strictly increasing version is A342517.
%Y A342518 The weakly decreasing version is A342519.
%Y A342518 A000041 counts partitions (strict: A000009).
%Y A342518 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342518 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342518 A045690 counts sets with maximum n with all adjacent elements y < 2x.
%Y A342518 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342518 A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
%Y A342518 A342098 counts (strict) partitions with all adjacent parts x > 2y.
%Y A342518 Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.
%K A342518 nonn
%O A342518 0,4
%A A342518 _Gus Wiseman_, Mar 20 2021