OFFSET
0,4
COMMENTS
Or, partitions whose length is a subset-sum of the parts.
EXAMPLE
The partition (3,2,1,1) has submultisets (3,1) or (2,1,1) with sum 4, so is counted under a(7).
The a(1) = 1 through a(8) = 15 partitions:
(1) (11) (21) (22) (32) (42) (52) (62)
(111) (211) (221) (321) (322) (332)
(1111) (311) (2211) (331) (431)
(2111) (3111) (421) (521)
(11111) (21111) (2221) (2222)
(111111) (3211) (3221)
(4111) (3311)
(22111) (4211)
(31111) (22211)
(211111) (32111)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n, 0, 10}]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
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Triangles:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 11 2023
STATUS
approved