cp's OEIS Frontend

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

%I A367395 #7 Nov 20 2023 08:19:04
%S A367395 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 A367395 68,78,92,107,124,143,166,192,220,254,291,335,382,439,499,572,649,741,
%U A367395 840,956,1080,1226,1383,1566,1762,1988,2235,2515,2822,3166,3547
%N A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.
%e A367395 The strict partition (5,3,2,1) has 4 = 3 + 1 so is counted under a(11).
%e A367395 The a(6) = 1 through a(17) = 7 strict partitions (A..E = 10..14):
%e A367395   321  421  521  621  721   821   921   A21   B21   C21    D21    E21
%e A367395                       4321  5321  6321  5431  6431  6531   7531   7631
%e A367395                                         7321  8321  7431   8431   8531
%e A367395                                                     9321   A321   9431
%e A367395                                                     54321  64321  B321
%e A367395                                                                   65321
%e A367395                                                                   74321
%t A367395 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,30}]
%Y A367395 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 A367395              sum-full  sum-free  comb-full comb-free semi-full semi-free
%Y A367395             -----------------------------------------------------------
%Y A367395 partitions:  A367212   A367213   A367218   A367219   A367394   A367398
%Y A367395 strict:      A367214   A367215   A367220   A367221   A367395*  A367399
%Y A367395 subsets:     A367216   A367217   A367222   A367223   A367396   A367400
%Y A367395 ranks:       A367224   A367225   A367226   A367227   A367397   A367401
%Y A367395 A000041 counts partitions, strict A000009.
%Y A367395 A002865 counts partitions whose length is a part, complement A229816.
%Y A367395 A088809/A093971 count twofold sum-full subsets.
%Y A367395 A236912 counts partitions containing no semi-sum, ranks A364461.
%Y A367395 A237113 counts partitions containing a semi-sum, ranks A364462.
%Y A367395 A237668 counts sum-full partitions, sum-free A237667.
%Y A367395 A366738 counts semi-sums of partitions, strict A366741.
%Y A367395 Triangles:
%Y A367395 A008284 counts partitions by length, strict A008289.
%Y A367395 A365541 counts subsets with a semi-sum k.
%Y A367395 A367404 counts partitions with a semi-sum k, strict A367405.
%Y A367395 Cf. A000700, A238628, A363225, A364272, A364534, A365661, A365925, A367410, A367411.
%K A367395 nonn
%O A367395 0,11
%A A367395 _Gus Wiseman_, Nov 19 2023