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A342517 Number of strict integer partitions of n with strictly increasing first quotients.

%I A342517 #11 Feb 16 2025 08:34:01
%S A342517 1,1,1,2,2,3,3,4,5,6,7,8,8,10,11,13,14,16,16,19,21,23,27,29,31,34,36,
%T A342517 40,43,47,49,53,56,59,66,71,75,81,86,89,97,104,110,119,123,132,143,
%U A342517 148,156,168,177,184,198,209,218,232,246,257,269,282,294
%N A342517 Number of strict integer partitions of n with strictly increasing first quotients.
%C A342517 Also the number of reversed strict partitions of n with strictly increasing first quotients.
%C A342517 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 A342517 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342517 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342517 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342517 The partition (14,8,5,3,2) has first quotients (4/7,5/8,3/5,2/3) so is not counted under a(32), even though the differences (-6,-3,-2,-1) are strictly increasing.
%e A342517 The a(1) = 1 through a(13) = 10 partitions (A..D = 10..13):
%e A342517   1   2   3    4    5    6    7    8     9     A     B     C     D
%e A342517           21   31   32   42   43   53    54    64    65    75    76
%e A342517                     41   51   52   62    63    73    74    84    85
%e A342517                               61   71    72    82    83    93    94
%e A342517                                    521   81    91    92    A2    A3
%e A342517                                          621   532   A1    B1    B2
%e A342517                                                721   632   732   C1
%e A342517                                                      821   921   643
%e A342517                                                                  832
%e A342517                                                                  A21
%t A342517 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342517 The version for differences instead of quotients is A179254.
%Y A342517 The version for chains of divisors is A342086 (non-strict: A057567).
%Y A342517 The non-strict ordered version is A342493.
%Y A342517 The non-strict version is A342498 (ranking: A342524).
%Y A342517 The weakly increasing version is A342516.
%Y A342517 The strictly decreasing version is A342518.
%Y A342517 A000041 counts partitions (strict: A000009).
%Y A342517 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342517 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342517 A045690 counts sets with maximum n with all adjacent elements y < 2x.
%Y A342517 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342517 A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
%Y A342517 A342098 counts (strict) partitions with all adjacent parts x > 2y.
%Y A342517 Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.
%K A342517 nonn
%O A342517 0,4
%A A342517 _Gus Wiseman_, Mar 20 2021