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A367220
Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.
24
1, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 7, 10, 11, 15, 17, 22, 25, 32, 37, 46, 53, 65, 75, 90, 105, 124, 143, 168, 193, 224, 258, 297, 340, 390, 446, 509, 580, 660, 751, 852, 967, 1095, 1240, 1401, 1584, 1786, 2015, 2269, 2554, 2869, 3226, 3617, 4056, 4541, 5084
OFFSET
0,6
COMMENTS
The non-strict version is A367218.
EXAMPLE
The a(3) = 1 through a(10) = 7 strict partitions:
(2,1) (3,1) (3,2) (4,2) (5,2) (6,2) (7,2) (8,2)
(4,1) (5,1) (6,1) (7,1) (8,1) (9,1)
(3,2,1) (4,2,1) (4,3,1) (4,3,2) (5,3,2)
(5,2,1) (5,3,1) (5,4,1)
(6,2,1) (6,3,1)
(7,2,1)
(4,3,2,1)
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], UnsameQ@@#&&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.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
A365046 counts combination-full subsets, differences of A364914.
Sequence in context: A241518 A372593 A125616 * A141472 A029034 A343941
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 14 2023
STATUS
approved