A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 26, 31, 38, 44, 58, 64, 81, 95, 116, 137, 166, 192, 233, 278, 330, 385, 459, 542, 636, 759, 879, 1038, 1211, 1418, 1656, 1942, 2242, 2618, 3029, 3535, 4060, 4735, 5429, 6299, 7231, 8346, 9556, 11031, 12593, 14482, 16525

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 partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

For parts instead of sums we have A034296, ranks A073491.

For all subset-sums we have A126796, ranks A325781, strict A188431.

The complement for parts instead of sums is A239955, ranks A073492.

The complement for all sub-sums is A365924, ranks A365830, strict A365831.

The complement is counted by A367403.

The strict case is A367410, complement A367411.

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, A365661, A365918, A365921.

Programs

Mathematica

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