login
A366753
Number of integer partitions of n without all different sums of two-element submultisets.
8
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753
OFFSET
0,11
EXAMPLE
The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
(3221) (32211) (4321) (33221) (4332) (43321)
(32221) (43211) (5331) (53221)
(322111) (322211) (5421) (53311)
(3221111) (43221) (54211)
(322221) (332221)
(332211) (432211)
(432111) (3222211)
(3222111) (3322111)
(32211111) (4321111)
(32221111)
(322111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Total/@Union[Subsets[#, {2}]]&]], {n, 0, 30}]
CROSSREFS
Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Sequence in context: A376654 A004657 A054075 * A062798 A327060 A270817
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 07 2023
STATUS
approved