A367403 Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.

0, 0, 0, 0, 0, 1, 2, 5, 9, 13, 22, 30, 46, 63, 91, 118, 167, 216, 290, 374, 490, 626, 810, 1022, 1297, 1628, 2051, 2551, 3176, 3929, 4845, 5963, 7311, 8932, 10892, 13227, 16035, 19395, 23397, 28156, 33803, 40523, 48439, 57832, 68876, 81903, 97212, 115198

Offset: 0

Author

Gus Wiseman, Nov 17 2023

Keywords

• nonn

Comments

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.

Examples

			The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  .  (311)  (411)   (331)    (422)     (441)
                        (3111)  (421)    (431)     (522)
                                (511)    (521)     (531)
                                (4111)   (611)     (621)
                                (31111)  (3311)    (711)
                                         (4211)    (4311)
                                         (5111)    (5211)
                                         (41111)   (6111)
                                         (311111)  (33111)
                                                   (42111)
                                                   (51111)
                                                   (411111)
                                                   (3111111)
		

Crossrefs

The complement for parts instead of sums is A034296, ranks A073491.

The complement for all sub-sums is A126796, ranks A325781, strict A188431.

For parts instead of sums we have A239955, ranks A073492.

For all subset-sums we have A365924, ranks A365830, strict A365831.

The complement is counted by A367402.

The strict case is A367411, complement A367410.

A000009 counts partitions covering an initial interval, ranks A055932.

A086971 counts semi-sums of prime indices.

A261036 counts complete partitions by maximum.

A276024 counts positive subset-sums of partitions, strict A284640.

Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365658.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,15}]