A367410 - OEIS

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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

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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.

LINKS

Table of n, a(n) for n=0..60.

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.

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

Sequence in context: A366385 A070172 A273353 * A259197 A309559 A130128

Adjacent sequences: A367407 A367408 A367409 * A367411 A367412 A367413

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 18 2023

STATUS

approved