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A367410
Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.
5
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
0,4
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}]; If[d=={}, {}, Range[Min@@d, Max@@d]]==Union[d])&]], {n, 0, 30}]
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.
Sequence in context: A366385 A070172 A273353 * A259197 A309559 A130128
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 18 2023
STATUS
approved