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

0,7

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.

Links

Table of n, a(n) for n=0..47.

Example

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)

Mathematica

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

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.

Sequence in context: A288347 A109094 A325647 * A049753 A331809 A126326

Adjacent sequences: A367400 A367401 A367402 * A367404 A367405 A367406

Keyword

nonn

Author

Gus Wiseman, Nov 17 2023

Status

approved