login
Number of integer partitions of n whose length is a semi-sum of the parts.
8

%I #11 Nov 20 2023 08:14:13

%S 0,0,1,0,1,1,3,3,6,7,14,15,25,30,46,54,80,97,139,169,229,282,382,461,

%T 607,746,962,1173,1499,1817,2302,2787,3467,4201,5216,6260,7702,9261,

%U 11294,13524,16418,19572,23658,28141,33756,40081,47949,56662,67493,79639

%N Number of integer partitions of n whose length is a semi-sum of the parts.

%C We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

%e For the partition y = (3,3,2,1) we have 4 = 3 + 1, so y is counted under a(9).

%e The a(2) = 1 through a(10) = 14 partitions:

%e (11) . (211) (221) (321) (421) (521) (621) (721)

%e (2211) (2221) (2222) (3222) (3322)

%e (3111) (3211) (3221) (3321) (3331)

%e (3311) (4221) (4222)

%e (32111) (4311) (4321)

%e (41111) (32211) (5221)

%e (42111) (5311)

%e (32221)

%e (33211)

%e (42211)

%e (43111)

%e (331111)

%e (421111)

%e (511111)

%t Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,10}]

%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 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 A365543 counts partitions with a subset-sum k, strict A365661.

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

%Y Cf. A000700, A088809, A093971, A126796, A238628, A304792, A363225, A364534, A365541, A365924, A367402.

%K nonn

%O 0,7

%A _Gus Wiseman_, Nov 19 2023