

A342516


Number of strict integer partitions of n with weakly increasing first quotients.


5



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, 42, 45, 48, 51, 58, 59, 63, 70, 76, 80, 88, 94, 98, 105, 113, 121, 129, 133, 143, 153, 159, 166, 183, 189, 195, 210, 221, 231, 248, 262, 273, 284, 298, 312
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OFFSET

0,4


COMMENTS

Also called logconcaveup 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

Table of n, a(n) for n=0..60.
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.


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 nonstrict ordered version is A342492.
The nonstrict version is A342497 (ranking: A342523).
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n  1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
Cf. A000005, A003114, A003242, A005117, A057567, A067824, A238710, A253249, A318991, A318992, A342528.
Sequence in context: A178503 A211275 A240542 * A325391 A179254 A304430
Adjacent sequences: A342513 A342514 A342515 * A342517 A342518 A342519


KEYWORD

nonn


AUTHOR

Gus Wiseman, Mar 20 2021


STATUS

approved



