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A367218
Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.
24
1, 1, 1, 2, 4, 6, 8, 13, 18, 26, 35, 50, 66, 92, 119, 160, 208, 275, 350, 457, 579, 742, 933, 1185, 1476, 1859, 2300, 2868, 3531, 4371, 5343, 6575, 8003, 9776, 11842, 14394, 17351, 20987, 25191, 30315, 36257, 43448, 51753, 61776, 73342, 87192, 103184, 122253, 144211
OFFSET
0,4
COMMENTS
The Heinz numbers of these partitions are given by A367226.
EXAMPLE
The partition (4,2,1) has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(7).
The a(1) = 1 through a(7) = 13 partitions:
(1) (11) (21) (22) (32) (42) (52)
(111) (31) (41) (51) (61)
(211) (221) (321) (322)
(1111) (311) (411) (331)
(2111) (2211) (421)
(11111) (3111) (511)
(21111) (2221)
(111111) (3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combs[Length[#], Union[#]]!={}&]], {n, 0, 15}]
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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A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A008284 counts partitions by length, strict A008289.
A240855 counts strict partitions whose length is a part, complement A240861.
A365046 counts combination-full subsets, differences of A364914.
Sequence in context: A240730 A376624 A039846 * A094092 A072791 A058320
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 14 2023
EXTENSIONS
a(31)-a(48) from Chai Wah Wu, Nov 15 2023
STATUS
approved