A366753 Number of integer partitions of n without all different sums of two-element submultisets.

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

Author

Gus Wiseman, Nov 07 2023

Keywords

• nonn

Examples

			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)
		

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.

Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

Programs

Mathematica

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