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A367399
Number of strict integer partitions of n whose length is not the sum of any two distinct parts.
8
1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 13, 15, 19, 22, 27, 31, 38, 43, 51, 59, 70, 79, 94, 107, 124, 143, 165, 188, 218, 248, 283, 324, 369, 419, 476, 540, 610, 691, 778, 878, 987, 1111, 1244, 1399, 1563, 1750, 1954, 2184, 2432, 2714, 3016, 3358, 3730, 4143
OFFSET
0,4
EXAMPLE
The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
(6) (7) (8) (9) (10) (11) (12) (13)
(4,2) (4,3) (5,3) (5,4) (6,4) (6,5) (7,5) (7,6)
(5,1) (5,2) (6,2) (6,3) (7,3) (7,4) (8,4) (8,5)
(6,1) (7,1) (7,2) (8,2) (8,3) (9,3) (9,4)
(4,3,1) (8,1) (9,1) (9,2) (10,2) (10,3)
(4,3,2) (5,3,2) (10,1) (11,1) (11,2)
(5,3,1) (5,4,1) (5,4,2) (5,4,3) (12,1)
(6,3,1) (6,3,2) (6,4,2) (6,4,3)
(6,4,1) (6,5,1) (6,5,2)
(7,3,1) (7,3,2) (7,4,2)
(7,4,1) (7,5,1)
(8,3,1) (8,3,2)
(5,4,2,1) (8,4,1)
(9,3,1)
(6,4,2,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n, 0, 15}]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
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A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A365924 counts incomplete partitions, strict A365831.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367403 counts partitions without covering semi-sums, strict A367411.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Sequence in context: A036823 A035575 A036816 * A113788 A018243 A127207
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2023
STATUS
approved