A367410 Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.

1, 1, 1, 2, 2, 3, 4, 4, 4, 6, 6, 7, 7, 8, 8, 11, 9, 11, 11, 12, 12, 15, 14, 15, 16, 16, 16, 19, 18, 19, 22, 21, 21, 24, 22, 25, 26, 26, 26, 30, 28, 29, 32, 31, 32, 37, 35, 36, 38, 39, 39, 43, 42, 43, 47, 46, 49, 51, 52, 51, 58

Offset: 0

Author

Gus Wiseman, Nov 18 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 = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is not counted under a(7).
The a(1) = 1 through a(9) = 6 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)
		

Crossrefs

For parts instead of sums we have A001227:

- non-strict A034296, ranks A073491

- complement A238007

- non-strict complement A239955, ranks A073492

The non-binary version is A188431:

- non-strict A126796, ranks A325781

- complement A365831

- non-strict complement A365924, ranks A365830

The non-strict version is A367402.

The non-strict complement is A367403.

The complement is counted by A367411.

A000009 counts partitions covering an initial interval, ranks A055932.

A046663 counts partitions w/o submultiset summing to k, strict A365663.

A365543 counts partitions w/ submultiset summing to k, strict A365661.

Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#,{2}]; If[d=={},{}, Range[Min@@d, Max@@d]]==Union[d])&]], {n,0,30}]