

A236912


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


49



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
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OFFSET

0,3


COMMENTS

These are partitions containing the sum of no 2element submultiset of the parts, a variation of binary sumfree partitions where parts cannot be reused, ranked by A364461. The complement is counted by A237113. The nonbinary version is A237667. For reusable parts we have A364345.  Gus Wiseman, Aug 09 2023


LINKS



FORMULA



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.
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 *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2}]]=={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 09 2023 *)


CROSSREFS

The (strict) version for linear combinations of parts is A364350.
These partitions have ranks A364461.


KEYWORD

nonn,changed


AUTHOR



EXTENSIONS



STATUS

approved



