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A367395
Number of strict integer partitions of n whose length is the sum of two distinct parts.
8
0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 5, 7, 8, 11, 13, 17, 19, 25, 28, 35, 41, 49, 57, 68, 78, 92, 107, 124, 143, 166, 192, 220, 254, 291, 335, 382, 439, 499, 572, 649, 741, 840, 956, 1080, 1226, 1383, 1566, 1762, 1988, 2235, 2515, 2822, 3166, 3547
OFFSET
0,11
EXAMPLE
The strict partition (5,3,2,1) has 4 = 3 + 1 so is counted under a(11).
The a(6) = 1 through a(17) = 7 strict partitions (A..E = 10..14):
321 421 521 621 721 821 921 A21 B21 C21 D21 E21
4321 5321 6321 5431 6431 6531 7531 7631
7321 8321 7431 8431 8531
9321 A321 9431
54321 64321 B321
65321
74321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#, {2}], Length[#]]&]], {n, 0, 30}]
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.
A088809/A093971 count twofold sum-full subsets.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
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: A210955 A051697 A240869 * A065308 A035680 A238220
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2023
STATUS
approved