OFFSET
0,4
COMMENTS
Also called log-concave-up strict partitions.
Also the number of reversed strict integer partitions of n with weakly increasing first quotients.
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).
LINKS
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
EXAMPLE
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.
The a(1) = 1 through a(13) = 11 partitions (A..D = 10..13):
1 2 3 4 5 6 7 8 9 A B C D
21 31 32 42 43 53 54 64 65 75 76
41 51 52 62 63 73 74 84 85
61 71 72 82 83 93 94
421 521 81 91 92 A2 A3
621 532 A1 B1 B2
721 632 732 C1
821 921 643
832
931
A21
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&LessEqual@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 30}]
CROSSREFS
The version for differences instead of quotients is A179255.
The non-strict ordered version is A342492.
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A000929 counts partitions with all adjacent parts x >= 2y.
A167865 counts strict chains of divisors > 1 summing to n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 20 2021
STATUS
approved