A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 4, 5, 8, 10, 14, 16, 23, 27, 35, 42, 52, 61, 75, 89, 106, 126, 149, 173, 204, 237, 274, 319, 369, 424, 490, 560, 642, 734, 838, 952, 1085, 1231, 1394, 1579, 1784, 2011, 2269, 2554, 2872, 3225, 3619, 4054, 4540, 5077, 5671, 6332

Offset

0,9

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..55.

Example

The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
The a(7) = 1 through a(13) = 10 partitions:
  (4,2,1)  (4,3,1)  (5,3,1)  (5,3,2)  (5,4,2)  (6,4,2)    (6,4,3)
           (5,2,1)  (6,2,1)  (5,4,1)  (6,3,2)  (6,5,1)    (6,5,2)
                             (6,3,1)  (6,4,1)  (7,3,2)    (7,4,2)
                             (7,2,1)  (7,3,1)  (7,4,1)    (7,5,1)
                                      (8,2,1)  (8,3,1)    (8,3,2)
                                               (9,2,1)    (8,4,1)
                                               (5,4,2,1)  (9,3,1)
                                               (6,3,2,1)  (10,2,1)
                                                          (6,4,2,1)
                                                          (7,3,2,1)

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 A238007:

- complement A001227

- non-strict complement A034296, ranks A073491

- non-strict A239955, ranks A073492

The non-strict version is A367403.

The non-strict complement is A367402.

The complement is counted by A367410.

The non-binary version is A365831:

- non-strict complement A126796, ranks A325781

- complement A188431

- non-strict A365924, ranks A365830

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: A308954 A053097 A331443 * A367684 A035946 A303939

Adjacent sequences: A367408 A367409 A367410 * A367412 A367413 A367414

Keyword

nonn

Author

Gus Wiseman, Nov 17 2023

Status

approved