A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.

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

Links

Table of n, a(n) for n=0..59.

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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partitions: A367212 A367213 A367218 A367219 A367394 A367398

strict: A367214 A367215 A367220 A367221 A367395* A367399

subsets: A367216 A367217 A367222 A367223 A367396 A367400

ranks: A367224 A367225 A367226 A367227 A367397 A367401

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.

Cf. A000700, A238628, A363225, A364272, A364534, A365661, A365925, A367410, A367411.

Sequence in context: A210955 A051697 A240869 * A065308 A035680 A238220

Adjacent sequences: A367392 A367393 A367394 * A367396 A367397 A367398

Keyword

nonn

Author

Gus Wiseman, Nov 19 2023

Status

approved