OFFSET
0,4
COMMENTS
Also called log-concave-down strict partitions.
Also the number of reversed strict partitions of n with weakly decreasing 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 strict partition (10,7,4,2,1) has first quotients (7/10,4/7,1/2,1/2) so is counted under a(24), even though the first differences (-3,-3,-2,-1) are weakly increasing.
The a(1) = 1 through a(13) = 14 strict 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
321 61 71 72 82 83 93 94
421 431 81 91 92 A2 A3
432 541 A1 B1 B2
531 631 542 543 C1
4321 641 642 652
731 651 742
741 751
831 841
5421 931
5431
6421
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&GreaterEqual@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 30}]
CROSSREFS
The non-strict ordered version is A069916.
The version for differences instead of quotients is A320382.
The weakly increasing version is A342516.
The strictly decreasing version is A342518.
A000005 counts constant partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A057567 counts strict chains of divisors with weakly increasing quotients.
A167865 counts strict chains of divisors > 1 summing to n.
A342528 counts compositions with alternately weakly increasing parts.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 20 2021
STATUS
approved