A342499 Number of integer partitions of n with strictly decreasing first quotients.

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 11, 14, 15, 18, 20, 23, 26, 31, 34, 39, 42, 45, 51, 58, 65, 70, 78, 83, 91, 102, 111, 122, 133, 145, 158, 170, 182, 202, 217, 231, 248, 268, 285, 307, 332, 354, 374, 404, 436, 468, 502, 537, 576, 618, 654, 694, 737, 782, 830

Offset: 0

Author

Gus Wiseman, Mar 17 2021

Keywords

• nonn

Comments

Also the number of reversed partitions of n with strictly 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).

Examples

			The partition (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)  (32)   (33)   (43)   (44)   (54)
                   (31)  (41)   (42)   (52)   (53)   (63)
                         (221)  (51)   (61)   (62)   (72)
                                (321)  (331)  (71)   (81)
                                              (332)  (432)
                                              (431)  (441)
                                                     (531)
                                                     (3321)
		

Links

• 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.

Crossrefs

The version for differences instead of quotients is A320470.

The ordered version is A342494.

The strictly increasing version is A342498.

The weakly decreasing version is A342513.

The strict case is A342518.

The Heinz numbers of these partitions are listed by A342525.

A000005 counts constant partitions.

A000009 counts strict partitions.

A000041 counts partitions.

A001055 counts factorizations.

A003238 counts chains of divisors summing to n - 1 (strict: A122651).

A074206 counts ordered factorizations.

A167865 counts strict chains of divisors > 1 summing to n.

A342096 counts partitions with adjacent x < 2y (strict: A342097).

A342098 counts partitions with adjacent parts x > 2y.

Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]