|
|
A237113
|
|
Number of partitions of n such that some part is a sum of two other parts.
|
|
65
|
|
|
0, 0, 0, 0, 1, 1, 3, 3, 8, 10, 17, 22, 37, 47, 71, 91, 133, 170, 236, 301, 408, 515, 686, 860, 1119, 1401, 1798, 2232, 2829, 3495, 4378, 5381, 6682, 8165, 10060, 12238, 14958, 18116, 22018, 26533, 32071, 38490, 46265, 55318, 66193, 78843, 93949, 111503, 132326
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
COMMENTS
|
These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1]. Thus, a(6) = 3.
The a(0) = 0 through a(9) = 10 partitions:
. . . . (211) (2111) (321) (3211) (422) (3321)
(2211) (22111) (431) (4221)
(21111) (211111) (3221) (4311)
(4211) (5211)
(22211) (32211)
(32111) (42111)
(221111) (222111)
(2111111) (321111)
(2211111)
(21111111)
(End)
|
|
MATHEMATICA
|
z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]], Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2}]]!={}&]], {n, 0, 30}] (* Gus Wiseman, Aug 09 2023 *)
|
|
CROSSREFS
|
These partitions have ranks A364462.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|