A236912 Number of partitions of n such that no part is a sum of two other parts.

1, 1, 2, 3, 4, 6, 8, 12, 14, 20, 25, 34, 40, 54, 64, 85, 98, 127, 149, 189, 219, 277, 316, 395, 456, 557, 638, 778, 889, 1070, 1226, 1461, 1667, 1978, 2250, 2645, 3019, 3521, 3997, 4652, 5267, 6093, 6909, 7943, 8982, 10291, 11609, 13251, 14947, 16984, 19104

Offset

0,3

Comments

These are partitions containing the sum of no 2-element submultiset of the parts, a variation of binary sum-free partitions where parts cannot be re-used, ranked by A364461. The complement is counted by A237113. The non-binary version is A237667. For re-usable parts we have A364345. - Gus Wiseman, Aug 09 2023

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A000041(n) - A237113(n).

Example

Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 11 - 3 = 8.
From Gus Wiseman, Aug 09 2023: (Start)
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (3111)    (421)      (521)
                                     (111111)  (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)
(End)

Mathematica

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2}]]=={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 09 2023 *)

Crossrefs

For subsets of {1..n} we have A085489, complement A088809.

The complement is counted by A237113, ranks A364462.

The non-binary version is A237667, ranks A364531.

The non-binary complement is A237668, ranks A364532.

The version with re-usable parts is A364345, ranks A364347.

The (strict) version for linear combinations of parts is A364350.

These partitions have ranks A364461.

The strict case is A364533, non-binary A364349.

The strict complement is A364670, with re-usable parts A363226.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A108917 counts knapsack partitions, ranks A299702.

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A007865, A151897, A275972, A325862, A326083, A363225, A363260, A364346, A364755.

Sequence in context: A028815 A014423 A101902 * A215966 A114312 A095041

Adjacent sequences: A236909 A236910 A236911 * A236913 A236914 A236915

Keyword

nonn

Author

Clark Kimberling, Feb 01 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 17 2023

Status

approved