A342498 Number of integer partitions of n with strictly increasing first quotients.

1, 1, 2, 2, 4, 4, 5, 6, 8, 9, 12, 12, 14, 16, 18, 20, 24, 26, 27, 30, 35, 37, 45, 47, 52, 56, 61, 65, 72, 77, 83, 90, 95, 99, 109, 117, 127, 135, 144, 151, 164, 172, 181, 197, 209, 222, 239, 249, 263, 280, 297, 310, 332, 349, 368, 391, 412, 433, 457, 480, 503

Offset: 0

Author

Gus Wiseman, Mar 17 2021

Keywords

• nonn

Comments

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

Examples

			The partition y = (13,7,2,1) has first quotients (7/13,2/7,1/2) so is not counted under a(23). However, the first differences (-6,-5,-1) are strictly increasing, so y is counted under A240027(23).
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)
                   (211)  (311)  (51)   (61)   (62)   (72)
                                 (411)  (322)  (71)   (81)
                                        (511)  (422)  (522)
                                               (521)  (621)
                                               (611)  (711)
                                                      (5211)
		

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 A240027.

The ordered version is A342493.

The weakly increasing version is A342497.

The strictly decreasing version is A342499.

The strict case is A342517.

The Heinz numbers of these partitions are A342524.

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],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]