A367395 - OEIS

%I #7 Nov 20 2023 08:19:04

%S 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,

%T 68,78,92,107,124,143,166,192,220,254,291,335,382,439,499,572,649,741,

%U 840,956,1080,1226,1383,1566,1762,1988,2235,2515,2822,3166,3547

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

%e The strict partition (5,3,2,1) has 4 = 3 + 1 so is counted under a(11).

%e The a(6) = 1 through a(17) = 7 strict partitions (A..E = 10..14):

%e   321  421  521  621  721   821   921   A21   B21   C21    D21    E21

%e                       4321  5321  6321  5431  6431  6531   7531   7631

%e                                         7321  8321  7431   8431   8531

%e                                                     9321   A321   9431

%e                                                     54321  64321  B321

%e                                                                   65321

%e                                                                   74321

%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,30}]

%Y 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.

%Y              sum-full  sum-free  comb-full comb-free semi-full semi-free

%Y             -----------------------------------------------------------

%Y partitions:  A367212   A367213   A367218   A367219   A367394   A367398

%Y strict:      A367214   A367215   A367220   A367221   A367395*  A367399

%Y subsets:     A367216   A367217   A367222   A367223   A367396   A367400

%Y ranks:       A367224   A367225   A367226   A367227   A367397   A367401

%Y A000041 counts partitions, strict A000009.

%Y A002865 counts partitions whose length is a part, complement A229816.

%Y A088809/A093971 count twofold sum-full subsets.

%Y A236912 counts partitions containing no semi-sum, ranks A364461.

%Y A237113 counts partitions containing a semi-sum, ranks A364462.

%Y A237668 counts sum-full partitions, sum-free A237667.

%Y A366738 counts semi-sums of partitions, strict A366741.

%Y Triangles:

%Y A008284 counts partitions by length, strict A008289.

%Y A365541 counts subsets with a semi-sum k.

%Y A367404 counts partitions with a semi-sum k, strict A367405.

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

%K nonn

%O 0,11

%A _Gus Wiseman_, Nov 19 2023